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Key takeaways

Successful project management depends on a team-wide understanding of roles and responsibilities. Using a RACI matrix to assign and define each role is a great way to keep a project on track and positioned for success.

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How Does a RACI Chart Help Project Managers?

Project managers use RACI charts to keep track of team roles and relay those responsibilities to the larger team. The matrix defines clear roles and responsibilities for individual team members across the various phases of the project, breaking each role down into four types of designation: those who are Responsible and Accountable for project deliverables, those who should be Consulted as work begins, and stakeholders who need to be Informed of ongoing progress, roadblocks, and updates. 

Read more: Project Management Phases

RACI Matrix Definitions 

Responsible.

The individual(s) with responsibility for the task or deliverable is typically responsible for developing and completing the project deliverables themselves. The responsible parties are typically hands-on team members who make direct contributions toward the completion of the project. The responsible team is comprised of the project’s “doers”, working hands-on to ensure that each deliverable is completed. 

Some examples of responsible parties are:

• Project Managers
• Business Analysts
• Graphic Designers
• Copywriters

Accountable

Accountable parties ensure accountability to project deadlines, and ultimately, accountability to project completion. This group frequently also falls under the informed category.

Some examples of accountable parties are:

• Product Owners
• Signature Authorities
• Business Owners
• Key Stakeholders

Consulted individuals’ opinions are crucial, and their feedback needs to be considered at every step of the game. These individuals provide guidance that is often a prerequisite to other project tasks, for example, providing legal guidance on a project throughout the process. If you are working on new product development or expansion, this could essentially be the entire organization.

Some examples of consulted parties are:

• Legal Experts
• Information Security and Cybersecurity Experts
• Compliance Consultants

Informed persons are those that need to stay in the loop of communication throughout the project. These individuals do not have to be consulted or be a part of the decision-making, but they should be made aware of all project updates. Typically, this party are business owners or stakeholders that are more interested in viewing the project at a 30,000-foot view.  Keep this group on your cc list for awareness of topics, decisions, and progress – that includes making them part of the initial project kickoff and project demos as optional attendees. This group often also falls under the accountable group.

Some examples of informed parties are:

• Project Committee Members
• External Stakeholders

Read more: DACI vs RACI Model Guide

Why Are RACI Roles Important?

RACI roles provide a sense of organization and clarity for teams that are looking to divide roles and keep team members accountable for their contributions. Considering that 27% of projects go over budget, for reasons like scope creep and lack of defined roles, RACI roles help position a project for success and avoid common pitfalls. 

Moreover, RACI roles help ensure that communication between all roles is ongoing. When you consider that nearly half of all project spending is at risk of being wasted due to a lack of effective team-based communication, it becomes all that more important to prioritize. Ultimately, teams who prioritize communication and well-defined roles are better off, and RACI roles help teams achieve that goal faster – while providing accountability for each team member’s unique contributions to the success of the project. 

Read More: Top 10 Main Causes of Project Failure

How to Create a RACI Matrix 

If you’re looking to implement a RACI matrix as part of your team’s project planning process, take these steps to create a RACI matrix.

Ensure that you have a thorough understanding of the project and its demands before outlining any further steps by communicating with key stakeholders and decision-makers.

Determine the list of key activities and deliverables from the director of program management or other leadership. 

Determine who is needed to be a part of the project or initiative.

Determine the project roles and responsible job titles and persons for each activity and deliverable.

Hold review sessions with key members of the team for alignment, and if you haven’t already, host a kickoff meeting with the entirety of the team and key stakeholders to unveil the matrix, address questions, and more. 

If the project has already started, it’s not too late to implement a RACI matrix.

• Outline the story. Using research from multiple sources, do a, b, c, and d.
• Utilize steps 2 and 3 (shown above). Ensure the right groups are assigned and engaged. 
• Hold a review session. Ensure that the team acknowledges and discusses the plan and the roles assigned.

Read more: 8 Factors That Lead to Successful Projec ts

Examples of a RACI Matrix

As shown above, a RACI matrix helps break down what roles individuals will play as work is carried out and to what extent they will be involved in the project overall. The horizontal axis represents each person on the project team and the vertical axis represents each task.

Each square of the matrix represents an individual, a task, and that individual’s role within the project, either responsible, accountable, consulted, or informed. In this situation, for example, the project manager is accountable for accessing risk, defining performance requirements, creating designs, executing construction, and approving construction work. However, they are only informed about approving construction work and defining functional and aesthetic needs.

Read more: Understanding Different Types of Stakeholders and Their Roles

Our FREE Downloadable RACI Matrix Template

Who creates the raci matrix.

The RACI matrix — sometimes called RACI model, RACI diagram, or simply just RAC — is created by the project manager at the start of the project as a key part of establishing the initial human resources planning for the project. Because miscommunication is a common threat to any project, RACI charts are a great asset to teams dealing with any type of project, from very simple projects to extremely complex ones. 

Common Mistakes When Creating a RACI Matrix

• Failure to plan ahead: Utilizing a RACI matrix should not be your first step in project planning. Having a fully assembled project team and at least a general idea of a task list and project plans is a better place to start before preparing a matrix.
• Working with too large a team: A RACI matrix is likely not the best bet for a large team, as it will make the matrix hard to understand and overly complex.
• Not communicating with the project team: A RACI matrix should help organize tasks and responsibilities that have already been introduced to the project team – no one likes to be blindsided. Be sure to host a kickoff meeting with the team first before creating a RACI matrix.

Frequently Asked Questions

Implementing a RACI matrix takes more than just a few emails and sporadic conversations – it takes consistent communication and planning. You should host a kickoff meeting to introduce the matrix to the team and make a plan to continue meeting at predetermined times throughout the project lifecycle. 

Here are a few more tips to keep in mind as you implement your RACI matrix within the team dynamic:

• Get everyone prepared. Send the document around to the meeting distribution as read-ahead material, requesting feedback if there are any major concerns. 
• Roll out each role for the team . During the meeting, conduct a review of the tasks and responsible parties. Do not rush through this review, but rather ensure enough time in your project kickoff for this important aspect. (Be certain to clarify the definitions of RACI to avoid ambiguity.)
• Consider changes and update accordingly. After the meeting, send out the notes documenting acceptance or updates to the RACI. In addition to sending out the notes, request any corrections within a reasonable yet defined timeframe. Clarify that if no changes are requested, each person is acknowledging their role and committing to the project tasks as outlined.
• Stay in touch. Consider a quick review with the entire team each quarter or every six months for longer projects to ensure it remains up-to-date and not simply another document in the repository but a relied-upon artifact.

As you implement the RACI matrix…

• Encourage teamwork and foster collaboration whenever possible.
• Don’t fear updates – make changes and adjustments as needed (but be sure to communicate those changes clearly to all parties).
• Earlier is better. Roll out your matrix plan to the team BEFORE you plan to implement it for the best results. 
• Have a clear-cut understanding of the project scope and how each role connects to the overall project goal.

For “Responsible” Parties:

• Make sure your project’s definition of Responsible is clear on who holds the “decider” role for the project or project phase’s completion, and what the dimensions of that responsibility will be.
• Ensure that all parties are aware of their role and responsibilities within the matrix.

For “Accountable” Parties: 

• When multiple Accountable team members must exist, use your definitions to make clear which individual is accountable for a given project element, and how that individual needs to interact with other Accountable team members.
• Ensure that there is only one “Accountable” party assigned per task.
• Be sure that the Accountable party has the authority and power to oversee the task as the accountable party.

For Consulted and Informed Parties: 

• Consulted parties are often high-level decision-makers with heavy schedules. Make sure you’re clear on their availability ahead of time.
• Similar to Consulted parties, Informed parties are often less hands-on and have less understanding of day-to-day project operations. As the project goes on, make sure to keep detailed notes to keep the Informed party up-to-date on key information. 
• Understand the ways that these parties like to communicate and create a plan to reach them early – whether that’s over phone calls, emails, video calls, or from within your project management system’s collaboration tools.
• Knowing the difference between who needs to be consulted versus informed can be a challenge if there is ambiguity about project roles. Consider what aspects of the project different team members need to know to do their jobs, and then bake those into your definitions.

RACI Matrix Pros & Cons

• Increased Engagement: RACI helps engage project participants in the project lifecycle. 
• Enhanced Project Planning: Project managers make project planning more organized, efficient, and detailed.
• Identifiable Improvement Opportunities: Areas of improvement are more easily identified.
• Easier Collaboration: Use of a RACI matrix creates a clear path for leadership to sign off on project steps, as project documentation in the RACI model is heavily emphasized.
• Better Communication: Improves overall group communication as a whole.
• Group Accountability: Assists groups, especially larger project teams, stay connected and accountable to their roles and project goals
• Limitations on Role Scope: The RACI model does not provide details on role scope, especially for responsible parties. These gaps in detail also affect other team roles, for example, another gap in a RACI is the determination of who is responsible for verifier and signatory.
• Limits on Task Details and Scope: While a RACI matrix can provide an overview of who is responsible for different tasks, it will not state what needs to be done.
• Not Aligned to the Agile Methodology: Project managers using an agile methodology like scrum may find it redundant since accountability, ownership, and ongoing communication is built into the scrum framework (i.e., product owner, scrum master, and daily standups with the team). Additionally, agile focuses on team-based delivery and accountability, while the RACI framework and alternatives focus on individual responsibility and autonomous accountability.

Read more: Top 10 Causes of Project Failure

Free RACI Matrix Templates

A number of project management software solutions include a native RACI matrix template. Here are just a few we’ve found:

Colorful RACI Chart Template

We love this template from Smartsheet because it’s colorful, thorough, and includes room for every party involved in the project. 

Pastel Colored RACI Matrix Template

This template from the Academy to Innovate HR is a great choice for project managers who want to organize their team roles with an easy-on-the-eyes chart that evolves beyond the simple spreadsheet. 

Simple RACI Chart from Clickup

These RACI templates from Clickup have enough variety to fit any of your project needs, but are simple enough for even beginner PMs to use.

Detailed RACI Matrix Template

This template is a great starter template for anyone looking to explore RACI charts in their project management strategy . As an added bonus – it comes with the RACI definitions already built in!

Excel-Based RACI Chart Template

Are you an Excel or Google Sheets user looking to take advantage of the RACI matrix? An Excel-formatted template from Project Management Docs can be just the solution for you. This template is a great template for users who want a chart that comes in a pre-formatted structure.

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A Comprehensive Project Management Guide for Everything RACI

By Kate Eby | July 15, 2016 (updated September 5, 2023)

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To ensure collaboration and project success, it is crucial for all project stakeholders to understand their roles and responsibilities and those of other project members. This is especially important when project teams are more complex due to their large size, involvement of distributed team members, or reliance on staff from multiple departments. 

RACI stands for Responsible Accountable Consulted Informed. While its origins are murky, the RACI matrix has been adopted by many organizations to associate roles with project deliverables. One Six Sigma tutorial describes RACI this way:

“Typically a task is associated with at least one role or in some cases multiple roles. This ‘Association’ of the role with a task can be divided into the following four association types:

• R esponsible
• A ccountable

The above four types of association of a role to a task represented in a simple task vs. role diagram or matrix is called RACI (pronounced ‘racey’) matrix. So basically the RACI matrix is a responsibility assignment matrix (RAM), designed to assign tasks, activities, responsibilities, accountability, decision making, support to team members of a process/project, and clarify expectations on the level of their participation.” Here is an example of a RACI matrix:

This guide will explain when to use RACI, why it works, guidelines to use it effectively, and the role it can play in effective project management in all industries from construction to healthcare. Lastly, we'll show you that once you've created your own matrix, you'll need a collaborative, real-time tool, like Smartsheet, to manage the rest of the project details — from start to finish.

The Four Association Types

RACI has four association types: responsible, accountable, consulted, and informed. Responsible roles produce deliverables; accountable roles check the deliverables; consulted roles advise on tasks; and informed roles are kept informed throughout these processes.

• Responsible: These roles are responsible for completing the task or deliverable. For example, if the responsibility role is a technical writer, this person may be responsible for writing online help files. A software developer wouldn’t write the help files, but might incorporate those files into the product, which would be defined as a different task.
• Accountable: This type of role has the final authority on (or is accountable for) the task’s completion. To take the previous example of a technical writer developing online help and a software developer incorporating the help files, a product manager might be responsible for ensuring that the files make it into the product.
• Consulted: This role functions as an adviser to a task. For example, a team may consult with a Scrum Master as a subject matter expert (SME). Consider advisers carefully, as having too many people in this role can stretch the task time and raise the risk of poor performance.
• Informed: Informed team members are kept up to date on task completion. Charting this role helps to illustrate dependencies among tasks and also ensures transparency into task status. It can be difficult to identify those who need to be informed, so consult various roles to determine who needs status updates. For example, the sales manager may require status updates because a customer has a special interest in feature development.

When to Use RACI

A RACI model is a useful tool for general project management. Use a RACI matrix when processes are stalled due to role confusion, or when role authority is not clearly outlined. We’ve outlined some common use cases for RACI below:  

• If the approval process is bogged down, it may be due to role confusion.
• If decisions are being overruled often and seemingly arbitrarily, this is also a situation that can benefit from clarifying roles.
• Another situation that frequently occurs is that many people perform the same analysis tasks. When tasks are not getting done, it may be that nobody knows who should be doing them.
• When the authority to perform tasks is not understood, it may be time to define roles and tasks, responsibilities, and authority.

Eliminating this sort of confusion and clarifying roles and tasks is the chief function of a RACI Matrix.

RACI is among the most popular models, but there are several alternatives, including:

• PACSI – Used in situations where multiple stakeholders can review and veto the work of a single accountable person. The roles include Performed, Accountable, Control, Suggested, and Informed.
• RAPID – Created by Bain & Company to clarify decision accountability through clear role and responsibility delineation. The roles include Recommend, Agree, Perform, Input, and Decide.

The Value of RACI

Implementing a RACI model helps you to involve the most qualified people in your projects. Project managers can use RACI to quickly develop charts that provide clarity to the team. Some major benefits of the RACI model are as follows:

• Eliminating role confusion.
• Preventing over-allocation of resources to one project and under-allocation to another.
• Clearly defining roles to all the people who fill them (clear understanding of expectations is key to a smooth project and reduces the need for conflict resolution later).
• Ensuring no task is overlooked when resources are allocated.
• Providing a fast and efficient way to re-allocate resources when there is turnover. New people can quickly identify their roles in a project and the roles of those with whom they must interact.

Finally, because the Informed category is given equal weight, the RACI Matrix encourages communication between roles. Communication is the key to clearly understanding expectations, which results in a smooth project.

RACI Matrices

A RACI matrix is a collection of all project activities associated with the people or roles responsible for each. Your matrix should include all project elements, including planning, testing, design, and support.

To create a RACI matrix, consider the following steps: 

• Decide How to Chart the Matrix: You can use any number of tools or templates, including a spreadsheet, whiteboard, or software solution.
• Identify the Project Tasks or Deliverables: Meet with key stakeholders to develop a list of project tasks. For this discussion, tasks include necessary activities, like meetings or events, as well as tangible deliverables, like features or products. Plot tasks across the X or Y axis of the matrix. For example, if you are charting a software project developed under Agile, the sprint demo meeting may be a required activity and should be included in the matrix as a task. Don’t forget to add maintenance of the RACI Matrix as its own task. The project manager usually maintains the RACI Matrix.
• Identify the Project Roles: Project roles are labeled across the other axis of the matrix. The project roles make the matrix more understandable and can be useful for adding any data you may have forgotten. As you identify roles, add any tasks that apply to those roles on the task axis. The task axis is also useful for identifying roles and clarifying resource allocation. This is a good time to assign names to roles as well — one name per role is optimal.
• Label the Intersections of the Axes: Where the X and Y axes intersect, label the intersection with an R, A, C , or I to finalize the matrix with who should be responsible, accountable, consulted, or informed on each task.

RACI Guidelines

While RACI matrices will differ by project, there are some broad guidelines that you should always follow. Above all, your matrix should encourage teamwork and inform all people of their roles and assignments. We’ve laid out additional best practices below: 

• Avoid multiple levels of oversight – one level is enough 
• Encourage teamwork
• Maintain chart fluidity – make changes as needed and let people know when things change
• Assign only one Accountable per task
• Ensure Accountable assignees have authority to ensure the task is complete
• Avoid too many consultants as they can take up too much time (waiting for answers, gathering input, etc.) while too few can damage performance, so try to find the "sweet spot”  
• Inform everyone with a role of their assignment

Finding the Right RACI Template

RACI templates save time and provide a starting point for building your chart. Choose a simple, customizable template to ensure that it is as useful as possible for all team and project types.

Some templates use the X axis for the tasks and Y for roles, and others do the opposite. If tasks exceed roles, as they generally do, it is easier to use the X axis for tasks and the Y for roles as you will be able to see the greatest number of roles per task at a glance on most computer monitors. On the other hand, you may find it easier to filter a chart based on role (for instance, filtering to show only the “I” roles for a given task) with the tasks on the X axis. In addition, most templates use some form of color-coding.

Regardless of which template you select, using one will eliminate a lot of basic work in constructing the chart and will free your time to define roles and tasks.

RACI Project Management

RACI project management focuses on creating and managing a RACI matrix to identify and resolve conflicts in roles and revise role categorization. You should approach analysis as a team and encourage all roles to provide feedback.

A RACI Matrix is analyzed vertically and horizontally. Here are some things to look for when reviewing the role axis (whether this is horizontal or vertical):

• If one role has too many responsibilities, should some responsibilities be reallocated or should more people be assigned to the role?
• If only one person is Accountable, is it reasonable to expect that person to make all decisions and could it threaten the project by creating a bottleneck?
• Here are some things to look for when reviewing the Task axis (whether this is horizontal or vertical):
• If there is a task with no one Responsible, should someone be assigned or should the task be eliminated?
• If there is a task where no one is Accountable, who has decision-making authority?
• If there are multiple people Accountable for a task, avoid conflicts by making a single person accountable. 
• If too many people must be consulted, evaluate whether one person can be assigned who talks with others involved.

It is common, during the life of a project, to have team members experience role confusion. A RACI matrix is useful for clearly identifying roles associated with a project and improving productivity, especially when you’re suffering from role confusion. Some signs of role confusion are:

• Concern over who makes decisions —Decision-makers are usually labeled as Accountable, but it may be that decisions are being made by the Responsible party. If that’s the case, the team should know who makes decisions in each situation.
• Finger pointing —If work is not completed on time, finger pointing may result. The key to avoiding this is knowing who is Responsible. 
• Poor resource allocation —The RACI Matrix should make resource allocation very clear, but sometimes a single task can be extremely taxing, leading to some issues concerning who should be doing what and when.
• Lack of action because of ineffective communications —If someone is not Informed, they may not know to perform a task.
• Too many consultations because the wrong people are consulted —The Consulted should be clearly labeled to avoid jeopardizing the project schedule.

It is the duty of a project manager to step in if role confusion is suspected and clarify roles to ensure that everyone is aware of expectations.

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What Is a RACI Chart? How to Use RACI to Assign Project Roles

It’s a fact: Complex projects make it easy for teams to lose track of tasks.

You might have an air-tight project plan and a stellar team to back it up. But if you’re not crystal clear about assignments—or even involvement—on a task level, confusion, crankiness, and even demotivation will creep into your project team.

Lucky for you, avoiding those issues is as simple as creating a RACI chart. 

In this article, we’ll explain what RACI stands for and how it’s used in project management. We’ll also share a few practical examples so you can see how to apply the RACI model to different types of projects.

What is a RACI chart?

Raci definitions explained, benefits of the raci model in project management, how to make a raci chart, raci rules and best practices.

• RACI chart examples

When to use or skip a RACI chart for your project

Common raci pitfalls and how to avoid them.

A RACI chart—also known as a responsibility assignment matrix —is a diagram used in project management to define team roles across 4 categories: Responsible , Accountable , Consulted , and Informed . It helps clarify who does the work, who calls the shots, whose opinion matters, and who needs to stay in the loop for each task, milestone, or decision.

A RACI chart enables you to visualize roles and responsibilities at a more granular level than simple resource assignments. That way team members and stakeholders know what’s expected of them so confusion doesn’t get in the way of project success.

RACI stands for Responsible , Accountable , Consulted , and Informed . Each letter in the acronym represents the level of ownership each person involved in a project will have on an individual deliverable. 

This simple chart gives you an at-a-glance view of RACI meanings and how many people to assign to each role in your RACI matrix .

R = Responsible

This team member does the work to complete the task. Every task needs at least one Responsible party, but it’s okay to assign more.

Examples of people you might assign to the Responsible role:

• Content writer
• Graphic designer
• UI/UX designer
• Software developer
• Business analyst
• QA specialist

A = Accountable

This person delegates work and is the last one to review the task or deliverable before it’s deemed complete. On some tasks, the Responsible party may also serve as the Accountable one. Just be sure you only have one Accountable person assigned to each task or deliverable. (Note: It might not be your project manager!)

Examples of people you might assign to the Accountable role:

• Project manager
• Product manager
• Department head

C = Consulted

Every deliverable is strengthened by review and consultation from more than one team member. Consulted parties are typically the people who provide input based on either how it will impact their future project work or their domain of expertise on the deliverable itself.‍

Examples of people you might assign to the Consulted role:

• Software architect
• Content editor
• Creative director
• Compliance officer
• Security specialist
• Legal counsel

I = Informed

Informed stakeholders simply need to be kept in the loop on project progress, rather than roped into the details of every deliverable.

Examples of people you might assign to the Informed role:

• Executive leadership
• External clients
• Team members assigned to dependent tasks
• Customer support team
• Administrative staff

Responsible vs Accountable meanings in RACI

The same person can be both Responsible and Accountable for a task in RACI—including a project manager. But they’re not one and the same. So what’s the difference?

• Responsible is a task-oriented designation that applies to the person (or people) actually completing the work. A whole team can be responsible for the execution of one task. ‍
• Accountable is an outcome-oriented designation that applies to a single person who reports on the work, whether in status updates or upon delivery. Being Accountable means you must answer for and/or sign off on the deliverable and deal with the consequences if it falls short of goals.

At its core, the RACI model helps you set clear expectations about project roles and responsibilities. That way you don’t have multiple people working on the same task or against one another because tasks weren’t clearly defined on the front end.

A RACI chart also encourages team members to take responsibility for their work—or defer to someone else when needed. Essentially, you’ll remove personal judgment and politics from your process and focus on your team’s ability to act responsibly within a framework you’ve created. Sounds pretty sweet, huh?

Building a RACI chart for your project is a relatively simple task. The hardest part is thinking through all the people involved in your project and what role makes the most sense for individuals at each stage of work.

You’ll want to map out a RACI chart for your project during the planning stage. This ensures responsibilities are clearly defined before work begins and gives you time to adjust to avoid any gaps or overlaps in assignments.

Here are the basic steps for making a RACI chart:

• List key project phases, tasks, and/or milestones in a column down the left side of your chart. You can get as detailed as you want, depending on the complexity of your project (and attention-span of your project team and stakeholders). 
• Enter the people involved in your project across the top row of your chart. Each individual should serve as the header of a single column. You can use names or job roles—whatever makes sense for your team and project.
• Go line by line down the chart, and assign each person across the row an R, A, C, or I to indicate the role they’ll play on that particular task.

Once your RACI chart is good to go, you can create a communication plan that aligns with the roles you’ve outlined for project teams and stakeholders.

Want to save time? Download our free RACI Excel template , or see how TeamGantt's built-in RACI feature works.

Using a RACI chart is a whole lot easier when you follow a few simple rules. Once your RACI chart is complete, review it to be sure it meets these criteria:

• Every task has at least one Responsible person.
• There’s one (and only one!) Accountable party assigned to each task to allow for clear decision-making.
• No team members are overloaded with too many Responsible tasks. You can use TeamGantt’s Workloads report to check availability across all your active projects.
• Every team member has a role on each task. (It’s not uncommon for some folks to be Informed on most tasks.)

These best practices can help you get the most out of RACI:

• Focus on project tasks, milestones, and decisions in the RACI chart. Avoid generic or administrative to-dos like team meetings or status reports .
• Align the tasks in your RACI chart with your project plan so there’s no confusion about details and due dates. (TeamGantt does this work for you by tying your RACI chart directly to your plan!)
• Keep RACI definitions close by because they can be tough to remember sometimes! ‍
• Assign the Responsible team members to tasks in TeamGantt .

RACI chart examples: Practical application in the real world

Let’s take a closer look at how you might put the RACI model to work on real-life projects. 

Producing a marketing handout

We’ll start with a simple example. Imagine you’re creating a RACI chart for a handout your marketing manager will distribute at an industry conference. 

Basic tasks for this project might include:

• Write project brief
• Create content
• Design handout
• Review first draft
• Update handout
• Approve final
• Send to printer

In this project example, we’ve assigned RACI roles to 7 key team members:

• Marketing manager
• Editorial director

Let’s zoom in on the RACI roles we mapped out for the Create content task example so you understand the why behind these assignments. 

• Responsible : The content writer is listed as Responsible for this task, so that’s who will actively work on content creation.
• Accountable : The editorial director is listed as Accountable for this task because that’s who is ultimately on the line for content quality and accuracy. Once the content is written, she’s the one who will review it to ensure it meets their company’s editorial standards.
• Consulted : The marketing manager is listed as Consulted . Since the marketing manager is the subject matter expert for the presentation, the writer can go to them for input or help filling in content gaps along the way.
• Informed : Several people have been assigned to the Informed role, though for different reasons. Since the Design handout task depends on this one, we want to make sure the writer keeps the creative director and designer informed on the status of content creation. The project manager and CMO are listed as Informed simply because they want to be kept in the loop about how work is progressing.

Developing a new software product

Now let’s look at a more complex project example. 

Developers who use an Agile workflow to tackle the job likely know what they need to do because there’s a constant stream of communication. But cross-functional departments and senior leaders might need more clarity. 

Here’s how you might map RACI roles to major tasks in a software development project , broken down by key tasks and RACI roles. (For the Informed assignments, we only listed people who need detailed progress updates to keep our example easier to read.)

Market Research

• Responsible : Business Analyst, Marketing Manager
• Accountable : Product Manager
• Consulted : Sales Representative, Customer Support
• Informed : Project Manager, Software Developers

Requirement Gathering

• Responsible : Business analyst
• Accountable : Product manager
• Consulted : UI/UX Designer, Software Architect
• Informed : Project manager, QA analysts

Design and Prototyping

• Responsible : UI/UX Designer
• Consulted : Business analyst, software developers
• Informed : Marketing manager, QA analysts

Software Development

• Responsible : Software Developers/Engineers
• Accountable : Software Architect
• Consulted : Product Manager, QA Analysts
• Informed : Project Manager, Technical Writer
• Responsible : QA Analysts/Engineers
• Accountable : Project manager
• Consulted : Software Developers, DevOps Engineer
• Informed : Product Manager, Technical Writer
• Responsible : DevOps Engineer
• Accountable : Project Manager
• Consulted : Software Developers, QA Analysts
• Informed : Product Manager, Customer Support

Maintenance

• Responsible : DevOps Engineer, Software Developers
• Consulted : QA Analysts, Technical Writer
• Responsible : DevOps Engineer, QA Analysts
• Consulted : Software Developers, Technical Writer

Marketing and Sales

• Responsible : Marketing Manager, Sales Representative
• Accountable : Marketing Manager
• Consulted : Product Manager, Customer Support

User Training

• Responsible : Customer Support Specialist
• Consulted : Technical Writer, UI/UX Designer
• Informed : All project team members

A RACI chart serves just about every project well. But it’s especially helpful when tasks require multiple resources, run concurrently, or depend on other tasks.

Here are a few scenarios when the RACI model is useful:

• The decision-making or approval process could hold up the project.
• There’s conflict about task ownership or decision-making.
• The project workload feels like it’s not distributed evenly.
• You experience turnover on a team and need to onboard someone quickly to a new role.

Of course, not all teams and projects are created equally. You might work with a team who just happens to communicate really well and stays on top of their own work. (Lucky you!) Or maybe your project is small enough that it would be silly to take the time to go through this exercise. 

In cases like these, don’t worry about taking the extra step of creating a RACI chart. Just be sure you have a clear plan in place to guide your team and project.

Further reading : How to Create a Realistic Project Plan: Templates & Examples

Now let’s walk through a few common mistakes that could hinder your RACI chart’s effectiveness.

Failing to get buy-in from your team and stakeholders

Creating a RACI chart in a vacuum is never a good idea. In a best-case scenario, you’d sit down with your team and stakeholders to walk through the role assignments on each task. But let’s be real: That’s not always possible.

Just be sure everyone represented has acknowledged and agreed to the roles and responsibilities you’ve laid out. More importantly, you want to check that your chart eliminates any further project confusion.

Setting it and forgetting it

It’s easy to build a RACI chart at the start of a project, then let it collect dust once the real work begins. But remember: This chart will defend you against mishaps that arise when you have too many cooks in the kitchen or a team member who thinks someone else is handling the work.

That’s why it’s important to keep these roles top of mind throughout a project’s life cycle. You can do this by reviewing RACI assignments for upcoming tasks in weekly status update meetings and making sure everyone involved in a project has easy access to the RACI chart. 

In TeamGantt, you can assign RACI roles directly in your project plan so they’re clearly visible as team members work their way to the finish line.

Overcomplicating stakeholder communication

If you have a lot of Consulted and Informed roles on your chart, make sure you have an easy and lightweight way to keep them informed. It could be as simple as making sure department heads and senior leaders have access to your project plan so they can follow progress along the way. 

Managing a project with external clients or stakeholders? Sharing a view-only link to your project in TeamGantt is a great option for looping in folks outside your organization.

Further reading: A Project Manager’s Guide to Effective Stakeholder Management

Keep teams in sync—and accountable—with TeamGantt

A RACI chart is a simple tool that makes projects easier to manage by creating less confusion and more accountability. But you’ve got more than roles and responsibilities to keep straight.

TeamGantt makes it easy to build a project plan your whole team can contribute to and collaborate on. Everything happens online, so you can stay on top of deadlines and monitor progress in real time.

Use our built-in RACI chart to assign roles and keep them visible from project start to finish, so everyone knows how they contribute to success.

Try TeamGantt’s Pro Manager plan free for 30 days!

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How to Make a Responsibility Assignment Matrix for a Project (Template Included)

The most important resource you’ll employ to deliver the project is people. They have to fit into the schedule and maintain the project budget. Defining what their roles and responsibilities are when executing tasks and delivering on the project goals is an important part of controlling the project.

How can you coordinate all the people who are involved in a project so they know what they’re doing and don’t block others from doing what they are assigned? Using a responsibility assignment matrix can help. An assignment matrix gives your project a team that gets things done.

What is a Responsibility Assignment Matrix in Project Management?

A responsibility assignment matrix (RAM) is a project management chart used to identify and define the various people and organizations and outline each of their roles in working on tasks or delivering a part of the project.

Project managers use an assignment matrix to clarify what cross-functional teams do within the boundaries of the project and its numerous processes. Sometimes a responsibility assignment matrix is required when responding to a request for proposal (RFP).

The responsibility assignment matrix can also be called a RACI matrix, which stands for responsible, accountable, consulted and informed.

• Responsible: Notes who is responsible for executing the task, which is then assigned to them.
• Accountable: Notes who has decision-making authority and how that power is delegated throughout the project team.
• Consulted: Notes who is able to offer insight into the task, from team members to stakeholders.
• Informed: Notes who is updated on what in terms of progress and performance, as well as when and how this information is disseminated.

This creates a map of connections between activities and project team members. Depending on the size of the project, there can be several assignment matrices used for various project levels.

Why Create a Responsibility Assignment Matrix?

The assignment matrix identifies what everyone on the team is responsible for, which means not only what their duties are, but how they participate in the project. Some will have defined tasks, others will offer help with work, while there are some who are designated as decision-makers. These groups all have an identity and function within the project to help guide it towards a successful end.

Clear communication leads to more efficient projects. An assignment matrix facilitates better communication between team members and provides transparency by creating a system to make sure everyone is updated and always on the same page. Belaboring communications can bog down a project with too many pointless meetings and confusing interactions in which people try to understand what they’re supposed to be doing. Using the responsibility assignment matrix helps, but having project management software that connects teams in real-time is ideal.

ProjectManager manages project information by allowing teams to attach files directly to tasks, and our unlimited file storage keeps important project documents at your fingertips anywhere, anytime. Commenting on tasks can save time and tagging others in the project team creates a communication process that avoids the pitfalls of redundancies or unnecessary meetings.

When Should a Responsibility Assignment Matrix Be Created?

The responsibility assignment matrix would be created at the start of the project. You’d want to have everyone on the project team aware of where they stand in terms of their involvement before they start executing tasks.

As much as its use is a preventative measure, it can be used prescriptively. If you’re deep into the project and things are not moving as planned, there could be communication gridlock. If team members are not in the loop, or misconstrue what they’re supposed to be doing, using a responsibility assignment matrix might untie up those knots in the communication channel.

If there’s a problem with leadership overruling suggestions on how to advance the project and this is seen as a problem, it’s likely that the roles and responsibilities of the project team need refining. The responsibility assignment matrix defines who has authority to make decisions and using it or revisiting can determine if the right people are in that position.

In fact, any of the definitions might need reexamining at any phase in the project. Perhaps tasks are falling behind schedule. This could be because team members aren’t aware of what tasks they own. Anytime a delay occurs, returning to the assignment matrix is a good first step, even if you went through the process as you should during the planning stage of the project.

How to Create a Responsibility Assignment Matrix

The actual making of a responsibility assignment matrix is not as difficult as getting everyone on board with what their roles and responsibilities are.

Therefore, you want to include your team in the process, get their input and eventually buy-in without spending too much time and energy on the process. Follow these steps to make sure everyone is in agreement and you’ll have a successful responsibility assignment.

• Identify all the participants involved in the project, from team members to stakeholders and everyone in between.
• List all deliverables associated with the project. Use a work breakdown structure to make sure you don’t miss any.
• Meet with team members on how to execute the tasks to create the deliverables. Every task needs to be discussed in terms of the team’s responsibility and authority.
• Draft the responsibility assignment matrix using a table with the project tasks listed on the left-hand column. Across the top add the name of everyone in the project.
• Where the tasks meet the project team member, assign whether they’re responsible, accountable, consulted or informed.
• When completed, share the responsibility assignment matrix with the project team and stakeholders and hold a meeting if necessary to make sure everyone understands their part in the project. If you’re working in a shared space, print out a copy and post it.

Free Responsibility Assignment Matrix Template

Using a RACI template is a shortcut that sets up your team and the project for success. ProjectManager is more than an award-winning software that organizes tasks, teams and projects to streamline work and boost productivity, it’s also the online hub for all things project management.

Among the hundreds of blog posts, guidebooks and tutorial videos are dozens of free templates that can help you through every phase of your project’s life cycle. Using our free RACI template will help you guide all the project teams better, allowing them to know where they stand in relation to the project and what their level of responsibility and accountability is.

Use it at the start of the project to avoid delays and untangle any communicative knots that are preventing the project from progressing as planned. To keep your project on track, download our free RACI template and get a head start on building a workable responsibility assignment matrix.

Best Practices

Using our free RACI template is a good start, but you have to make sure you fill it in correctly. A responsibility assignment matrix is only as good as the effort put into creating it. Here are some best practices to apply when you’re in the process of building your assignment matrix.

• Involve the team: They’re the ones who will be executing the work. You want their input and buy-in to avoid any costly mistakes or time-consuming questions about what wasn’t made clear at the beginning of the project.
• Identify every single task: Identify all the tasks required to reach your final deliverable. Once you have that thorough list make sure that there is only one person on the team who is accountable.
• Update your RACI regularly: Make sure that each new one is clearly marked as the most current version and is distributed to everyone on the team. There will be times when you’ll want to revisit the responsibility assignment matrix or changes in personnel will require an edit.
• Share responsibility viably: One person shouldn’t have to shoulder the bulk of the responsibilities for the project and you want to give authority throughout the project team and not just among the very top management team.
• Optimize tasks: Managers can use the RACI matrix to see if too many team members have been assigned to a task. Maybe these workers could be spread out for greater productivity. There could be too many people listed as consulted, which slows down the process. The assignment matrix is endlessly useful.

How ProjectManager Helps You Manage Projects Better

ProjectManager is a cloud-based tool that connects everyone in real-time to facilitate planning, monitoring and reporting on the project. It works to give everyone on the project team a job and the knowledge as to where they have authority and when to consult others, as well as defining the reporting process.

Let’s look at the people who are responsible, for example, the team who execute the project. Once invited into the software, you can share the project plan, assign them tasks, add detailed direction, add a deadline and tag for priority and more. The teams can then collaborate by attaching files and images to the tasks and commenting in real-time to work better together.

Those who need to stay informed of the project can do so by also getting invited into the project and sharing plans and schedules with them. Stakeholders can stay updated with reporting features that can generate reports on project variance, cost, time and more with one click. Then share them as a PDF. Reports can even be quickly filtered to zero in on the data stakeholders are interested in.

The responsibility assignment matrix can help you reallocate your resources when things aren’t progressing as planned. Use our software to get further insight. The resource management features include a workload chart that’s color-coded so it’s easy to see who has too many tasks and who can take on more work. Then you can simply reallocate those resources from the workload page to help your team work more productively.

ProjectManager gets you organized, keeps your team focused on their tasks and stakeholders in the loop. Gain efficiencies throughout every aspect of your project’s life cycle with an online Gantt chart to schedule work and kanban boards, a visual workflow feature that provides transparency into production. All that and it’s on a collaborative platform to keep everyone connected. Try ProjectManager today for free.

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Complete RACI Chart Guide 2024: Definition, Examples and Templates

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Project managers are tasked with ensuring that every team member knows what's expected of them. However, critical information surrounding roles and responsibilities can sometimes fall through the cracks. Fortunately, there's a tool that can help define project roles and associated responsibilities: the RACI chart.

Last Updated: 18 Apr'24 2024-04-18T23:46:16+00:00

All our content is written fully by humans; we do not publish AI writing. Learn more here.

We’re all familiar with Gantt charts and kanban boards that help us plan, assign tasks and monitor projects. However, the RACI chart, which can be found in some of the best project management software like monday.com and Asana, helps make project management easier by letting managers assign project responsibilities and roles.

Key Takeaways: RACI Chart

• RACI charts offer a simple way to introduce more organization and accountability to your projects.
• Most leading project management software has templates for RACI charts, which makes them easy to implement.
• The RACI matrix is suitable for most (but not all) types of projects or project methodologies.

If you’re a project manager looking for an easy-to-use tool that will help you define roles within your teams and create an extra layer of accountability, you’ll want to stick around. In this guide, our expert project managers will tell you about the benefits of RACI charts, how they work, and how you can create a RACI chart and incorporate it into your project workflow.

Updated to reflect changes in the naming of Asana’s pricing plans.

What Is a RACI Chart?

You’re probably wondering what a RACI (responsible, accountable, consulted, informed) chart is. A RACI chart — also known as the RACI matrix or a responsibility assignment matrix — is a visual tool that displays project roles and responsibilities of team members for specific tasks.

With so many moving parts in a project, every team member must know what’s expected of them. While Gantt charts, kanban boards and lists can help show who’s doing what, they aren’t as good as a RACI matrix when it comes to visualizing each team member’s role.

No matter what project management software you use, RACI charts are relatively easy to set up and read, and will quickly become a vital part of your project management process.

What Are the 4 Components of RACI Charts?

On the surface, the RACI model may seem complex, but it’s relatively simple. The RACI model uses four categories, and each project team member falls into one of the categories. The components of RACI are responsible, accountable, consulted and informed.

If you have been marked as responsible, you’re responsible for getting the project task done. If you’re tagged as an accountable person, you’re a decision-maker who delegates project tasks and delivers the final product. A consulted member is a subject-matter expert who provides feedback and advice, while informed members are just being kept in the loop.

What Is the Difference Between Responsible and Accountable in RACI?

A big issue some teams have regarding RACI charts is understanding the difference between two core RACI components — namely, the responsible and accountable tags.

The difference between responsible and accountable members is not complex. A team member marked as responsible is responsible for a particular task and ensuring their job is complete. It’s worth noting that there’s no limit on how many team members can be marked as responsible.

An individual marked “accountable” delegates and reviews the work the responsible person performs before the task gets marked as finished.

What Are RACI Chart Rules?

If you want your projects to run smoothly and not leave anyone out or overload one person with too many tasks, you should follow the four accepted RACI matrix rules.

Rule 1: At Least One Person Responsible

The first rule states that you should ensure every job has at least one team member marked as responsible . Remember, there’s no limit on how many team members can be marked as responsible.

Rule 2: Only One Decision Maker

The second rule is making sure that every task has only one accountable person for decision-making. 

Rule 3: Don’t Overload

Rule three ensures that no team members are overloaded with too many responsible tasks.

Rule 4: Each Member With At Least One Role

Rule four states that each team member should have at least one role in each job , even if they are just an informed participant. 

When Should You Create a RACI Matrix?

Not every project needs a RACI chart. However, should you find yourself in a situation where team members must know what’s expected of them, it’ll be worth taking the time to create a RACI chart. Of course, you’ll still need kanban boards and other tools to track project progress, but a RACI chart can be a part of project success.

You should create a RACI chart if:

• Team members need clarification on their roles. RACI charts clearly show who is responsible for tasks and deliverables.
• You want to streamline communication. A RACI chart shows the team who’s in charge of what at any given time, making it easier to reach out to the right person for advice or feedback.
• You want to spread out work evenly. Project managers can see at a glance if a team member has been assigned too many jobs.
• You want accountability at all levels. While one team member will be designated as accountable, the RACI matrix promotes accountability at all levels, thanks to clearly defined project roles.

Many more situations warrant using a RACI chart, including not wanting to overlook essential tasks and preventing project delays due to team members not knowing their roles and responsibilities. Only you can decide if a RACI chart will help you in your current and future project work.

Does RACI Work With Other Project Management Methodologies?

While RACI charts work well for most small- to medium-sized projects, they’re pretty redundant for projects without many stakeholders and those using scrum and Agile methods, as roles and responsibilities are often built into those methodologies. When used in complex projects, RACI charts can become quite messy if you have a large team, so keep this in mind.

Of course, everyone is different. Some project managers like using the RACI method regardless of how complex a project is, so while your project might not need a RACI chart, some managers may still find them beneficial to keep track of staff and their roles.

What Are the Benefits of Using a RACI Chart?

In most cases, if you take the time to create a RACI chart, you’ll reap the rewards that it can bring. When a team is on the same page, it can help lead to events and results that will please everyone. Here are a few examples of the benefits of using a RACI Chart.

You Can Plan Ahead

If you use a RACI chart, you’ll be able to plan each individual’s roles in a project before you even start working. As a resource management tool, a RACI chart can be hard to beat. Letting team members see what’s expected of them ahead of time will help set them up for success during each phase of the project.

There Will Be an Increase in Productivity

Using a RACI chart will likely increase the productivity of the people involved in your project. When RACI roles and responsibilities are clearly defined, team members can carry out the project plan with minimal fuss.

The RACI Model Promotes Organization and Communication

Organization is critical when it comes to managing projects. Fortunately, RACI charts add an extra layer of organization to a project by helping employees know their roles on any project. As a result, team members can also see who they need to reach out to for help or information.

What Are the Limitations of a RACI Chart?

So far, it sounds like using the RACI model would be a no-brainer, and for many teams, it is. However, it’s also important to know that RACI charts have limitations as well as benefits. So let’s look at how a RACI chart might hold you back.

RACI Roles Are Inflexible

The roles that can be assigned in a RACI chart are straightforward, but they do not convey an employee’s skill set or how knowledgeable they are about the task they have been assigned. The RACI roles also may not reflect what your stakeholders and team members actually do.

Deadlines Aren’t a Thing

RACI charts are good at conveying who is doing what, but they can’t share timelines with your responsible parties, informed or consulted groups, or accountable employees. Small teams may get away without using other project management tools, but most will need to use a RACI chart in conjunction with another tool, like a Gantt chart.

RACI Charts Do Not Define Tasks

A RACI chart will show team members who are responsible and accountable for a particular task. However, the charts do not give specific details about the job. Therefore, you may need to use another collaboration tool to list the details of each task.

Best Project Management Software With RACI Chart Templates

If you would still like to use RACI charts after reading about their limitations, you have options. You can easily create a RACI chart in Excel or Google Sheets, or you can use dedicated software like monday.com, Asana or ClickUp (check out our ClickUp Review ).

Fortunately, you don’t need mad skills to create RACI charts. Most mainstream project management tools provide a RACI chart template for you to use. Here, we’ll show you how to get your hands on the templates for monday.com , Asana and ClickUp .

1. monday.com

Creating a RACI chart on monday.com is as easy as heading to monday.com’s template generator . This page allows you to play around with a demo RACI matrix before you install the template. The user-friendly RACI template is available even with monday.com’s free plan . You can find out more about monday.com in our monday.com review .

• Maximum users: 2
• Minimum users: 3
• Enterprise-level features.

Asana also offers RACI chart templates. To see all the available templates, head over to Asana’s RACI matrix template . Asana offers easy-to-use RACI charts that fit many different types of projects. You can also create your own. However, to use the templates, you need to be a Starter plan member. Find out more about Asana’s Starter plan in our Asana review .

• Up to 10 users
• Price is per user. unlimited users, expanded features
• Price is per user. unlimited users, even more features
• Custom pricing, advanced security features

ClickUp is another project management platform we highly recommend (it made our list of the best free project management software ) for those who want to use RACI charts. Like monday.com and Asana, ClickUp has a friendly template landing page with lots of information about their template . ClickUp’s charts are colorful and easy to use. Read more about it in our ClickUp review .

• Basic functionality with some limitations
• All prices per user

How to Create a RACI Matrix 

Now that we have discussed the RACI matrix, we will show you how easy it is to create a RACI chart for your next project. We will be using monday.com for this example.

The first step you need to take is one that will take you to monday.com’s RACI template . From here, all you need to do is click the “use template button . ”

As soon as you hit the “template” button in step one, you’ll be whisked away to your monday.com account and presented with your new RACI chart . If you navigate away from the chart and need to find it, you see it listed in the left-hand column .

Now you need to populate the RACI chart. A project manager needs to add tasks; assign the responsible, accountable, consulted and informed tags to the project team; set a due date and mark where you are in the process. Then you need to update the chart if anything changes.

What Are the Best RACI Matrix Alternatives?

We like RACI charts, but know they’re not for everyone. We want to quickly cover some other responsibility assignment matrix options with you, as one might fit your needs better.

RASCI charts are very similar to RACI charts, but, as you can see, there’s an extra letter in the acronym. The “S” in RASCI stands for “supportive.” Team members in a supporting role help a responsible party get the work done.

DACI (driver, approver, contributor, informed) is a matrix designed for decision-making processes. The matrix clarifies who your lead decision-maker is for specific projects or tasks. If your team struggles to call the shots due to internal conflict, the DACI model might work for you.

The CARS model (communicate, approve, responsible, support) is very similar to the RACI model. The most significant difference is that this model relies on support roles, much like RASCI. 

The RAS model (responsible, approve, support) is a simplified version of CARS; however, this model eliminates external stakeholders as there is no room for consulting members outside the team.

The CLAM (contribute, lead, approve, monitor) model is nearly identical to the RACI model; it just has friendlier terminology. Contributors refer to those who are consulted and who do the work. Monitors replace the informed role, while lead is the same as the accountable tag. Finally, members dubbed “approve” are the decision-makers in the CLAM method.

Final Thoughts

The RACI matrix is a simple tool, but you should not underestimate its value. When a RACI chart is used correctly, project managers and team members will know where they stand and who they need to talk to about the task they’re completing. If you want your projects to have added layers of organization and accountability, consider using a RACI chart.

Have you used a RACI chart before? Did it help or hinder your team with project phases of a complex project? Do you plan to give the RACI matrix a try now? Please let us know in the comment section, and thanks so much for reading.

The RACI acronym stands for responsible, accountable, consulted, informed.

Even though the RACI matrix is quite simple, it can play an important role in bringing accountability to project teams.

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Your guide to RACI charts, with examples

Can you identify exactly who’s doing what by when for each task, milestone, and deliverable in your project? If not, you might need a RACI chart.

RACI is an acronym to help teams clarify project roles and figure out who the responsible party is for any given task. Whether you've never heard of RACI before or you’re considering creating a RACI chart for your next project, here’s everything you need to know about how to create and use these charts.

What is a RACI chart?

Responsible.  This person is directly in charge of the work. There should only ever be one Responsible role per task so you know who to go to with questions or updates. If a task has more than one Responsible person, you can lose clarity and cause confusion. Instead, aim to add additional collaborators as some of the other RACI roles, which can have more than one person.

Accountable.  The Accountable person is responsible for overseeing overall task completion, though they may not be the person actually doing the work. There are two ways to assign an Accountable role. Sometimes, the Accountable is the project manager (or even the Responsible, though in that case the person is taking on two different roles during the task workflow). In these cases, the Accountable is responsible for making sure all of the work gets done. In other cases, the Accountable is a senior leader or executive who is responsible for approving the work before it’s considered complete. Like the Responsible role, there should only ever be one Accountable.

Consulted.  This will be the person or people who should review and sign off on the work before it’s delivered. There may be multiple Consulted roles for each task,  project milestone , or deliverable.

Informed.  This is the person or group of people who are informed about the progress and completion of work. They probably are not involved in any other aspect of the deliverable.

When should I create RACI charts?

RACI charts are a helpful way to track each stakeholder’s role for a task, milestone, or deliverable—especially if you’re managing a complex project with many decision makers and subject matter experts. With a RACI chart, you can prevent poor decision making and avoid roadblocks in the approvals process that could impact overall project success.

These charts, while different from PERT charts , are especially useful if your stakeholders may be taking on different roles throughout the project. For example, there could be a stakeholder who is Responsible on one deliverable but Informed on another. With a RACI chart, you can clearly outline these details and make sure everyone knows who’s responsible for what.

Example of a RACI chart

To build a RACI chart, list every task, milestone, or deliverable for your project. Then, identify who the Responsible, Accountable, Consulted, and Informed team members are for each one.

Let’s say you’re updating the homepage on your website.  Project stakeholders  include:

Head of website

Web developer

You want to create a RACI chart for five tasks and deliverables:

Update homepage CTAs

Update customer story on homepage

Revamp website design

Improve homepage loading speed

Update homepage design

The RACI chart would look like:

Responsible: Copywriter

Accountable: Web developer

Consulted: Head of website

Informed: Designer

Revamp video on homepage

Responsible: Designer

Informed: Copywriter

Responsible: Web developer

Informed: Copywriter & Designer

Pros and cons of RACI charts

Ultimately, the question is: should you create a RACI chart? While RACI charts are a useful tool to identify project responsibilities, they can get a little cumbersome over the lifecycle of a project. Here are the pros and cons of creating a RACI chart for your team’s work:

The benefits of RACI charts

Clear project roles and responsibilities can help your team move fast and reduce confusion about who’s working on what. With a RACI chart, you can ensure you don’t have two team members working on the same thing. As a result, you’ll have an easier time  collaborating  with your team.

RACI charts are also particularly helpful when the decision-making process is split between tasks. There might be scenarios where the Informed on one task or milestone is the Responsible or Consulted on another—in order to have that clearly defined, it’s helpful to track this work in a RACI chart.

RACI chart pitfalls (and how to avoid them)

RACI models focus on the granular, instead of capturing work at the project level. You might know who the Consulted is on a particular task—which is helpful—but knowing that doesn’t help you understand how various stakeholders interact with the broader project work.

Additionally, if you attempt to write out each task and each role, your RACI chart can get bulky. Worse, if your project changes in some way, your RACI chart would immediately become outdated. That can make it hard for you to gain real-time clarity about where each task is in your project workflow.

RACI charts are limited because they aren’t able to adapt to your project needs in real time. In order to establish clear expectations and eliminate confusion on the project level, you need a  project management tool .

Take your RACI chart to the next level

With project management software, every task has an assignee—that’s the Responsible. You can see work on the project level, so the Accountable and Informed don’t have to check in via email or status meetings. And, for any approvals you need from your Consulted, you can track reviews and approvals in one place. That way, your entire RACI team has a central source of truth for all of the work being done.

Instead of having your RACI chart separate from where the work is happening, project management tools capture the topic, assignee, and other important information like the task due date or relative importance. That way, your entire project team has visibility into who’s doing what by when—and you’re not relying on a single person to manage and update your RACI chart. Project management tools update in real time, so you can see exactly where you are in the approval process.

Track who’s doing what by when

Clear team roles and responsibilities help you hit your deliverables on time. Tracking different and complex stakeholder responsibilities in a RACI chart can help you do that—but RACI charts are just the beginning. Learn more about  work management , and how your team can benefit.

Related resources

What is stakeholder analysis and why is it important?

How Asana uses work management to optimize resource planning

Understanding dependencies in project management

Program manager vs. project manager: Key differences to know

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What is a RACI chart and how to use it (with template)

Product work is complex. While there are frameworks and best practices, due to its nature, you can only apply the same process in some places.

One can take the best practices, analyze your context, and then define a way of working and responsibilities based on your specific situation. It depends on your company, the kind of products you are working on, and the overall philosophy and mindset you are dealing with.

There are no two situations where teams perform product work similarly.

The most overlaps I have come across are between the product manager and the UX role, and research performed by NN Group also holds evidence of this fact.

In this article, we’ll discuss where a responsibility assignment matrix, aka a RACI chart, can be helpful. RACI charts can help you align with your team and stakeholders on roles and responsibilities, while also improving communication and decision-making.

What is a RACI chart?

The RACI chart, or responsibility assignment matrix, is a tool that helps to communicate and clarify the roles and responsibilities of people working together. In product management, it adds support for alignment and communication in the product development process:

Responsible

This team member is the one responsible for performing the task. Each task needs at least one responsible person, but can also have more.

Accountable

This team member is the one who is ultimately accountable for the outcome and success of the task or deliverable. They may either perform the task themselves (in which case they are also responsible), or they may delegate the work to someone else.

You should only have one accountable team member per task.

You can have as many consulted stakeholders as you want. These team members and stakeholders are the ones whose input is required to complete the task. They provide information from their expertise or how the task result may impact their work.

These team members should stay in the loop. If you were to think of the power/interest grid for stakeholder management , you would consider stakeholders with low power and low interest in this category.

Applying the RACI chart to the product development process

You can apply the RACI chart to the different stages of the product development process. We will use the 4D methodology (discover, design, develop, and deliver) to showcase how the responsibilities can be split between product management, ux design, development team, and product marketing.

Depending on your context, you may also include a delivery manager, who will take over some of the responsibilities of the product manager:

Let’s break down the graphic above by defining the 4D methodology.

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The product manager is responsible for researching and understanding the target market’s needs and stakeholders. They are accountable for defining the project’s scope, allocating the right resources, and ensuring the research is done correctly.

The UX designer and development team are consulted for input and feedback on the user experience and technical feasibility.

Product marketing will be informed of the findings.

The UX designer is responsible for the user experience, while the architect/technical lead is responsible for the technical design.

The product manager is accountable for ensuring that the design meets the requirements defined in the discovery stage, that the design is feasible, and that resources are allocated appropriately.

The development team will be consulted for input and feedback on the technical feasibility.

Product marketing is merely informed at this stage as well.

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The development team is responsible for implementing the design and creating the product.

The product manager is accountable for ensuring that the product is developed on time and within budget and for managing any issues that arise during development.

The UX designer should be consulted for input and feedback on the user experience and any issues identified.

Product marketing is informed of the progress and status.

Product marketing is responsible for promoting and launching the product and informs of the product’s readiness for launch.

The product manager is responsible for product delivery to the end user or client. They are accountable for ensuring the product meets the requirements defined in the discovery stage, that the product is ready to be delivered, and that any final issues get solved.

The development team is consulted for input and feedback on the technical feasibility and any issues identified.

UX design is informed about the progress and status.

Generally, the person accountable for a given stage will be responsible for leading it and making the final decisions, while others are consulted and informed as appropriate.

Benefits of using a RACI Chart

A RACI chart is a great tool to help product teams organize and work together. Here are a few benefits:

RACI helps clearly define roles and responsibilities for different tasks and decisions, reducing confusion and ensuring that everyone knows what is expected of them. This applies both within the team and outside of it towards the organization.

Accountability

By clearly defining who is accountable for different tasks and decisions, RACI can help to ensure that everyone takes ownership of their responsibilities and is accountable for their actions.

Additionally, defining clear roles and responsibilities helps avoid double work and waste.

Consistency

RACI can ensure that tasks and decisions are handled consistently across the product management process, which can improve the overall quality and effectiveness of the process.

By clearly defining roles and responsibilities, RACI can help to streamline the product management process and make it more efficient, ultimately saving time and resources.

Improved communication

RACI can ensure that all stakeholders are kept informed about the progress of the product management process and any changes made, which can help improve communication and reduce misunderstandings.

Common pitfalls to avoid

Here are a few pitfalls to look after when considering working with the RACI Chart:

Forgetting about it and not updating it regularly

The RACI Chart is a living document — it evolves with your team or product. It is important to revisit it every time you experience a change. Even if no changes occur, you should review it once a quarter.

You should revisit the RACI chart anytime a change happens:

• A new member is joining the team
• A member is leaving the team
• There’s a new stakeholder for the product
• Change in the way of working
• The product enters a new stage in the life cycle

Not utilizing its potential for transparency

The RACI chart is a handy communication tool. Creating a RACI chart and shoving it into a folder or a drawer will not help you in any way.

Since it is a tool meant to increase alignment and transparency, you should ensure that every involved party is aware of it and has easy access. It is also beneficial when onboarding new team members or stakeholders.

Overlapping roles

When creating or maintaining your RACI chart, pay attention to how the responsibilities are split and that there are no overlapping roles. You should, as a team, clearly define who holds the responsibility and avoid having it split between too many team members, as this may cause unnecessary overhead.

Unbalanced responsibilities

Delivering successful products is a marathon, not a sprint. It is also vital to maintain a sustainable pace within the team, which also applies to the workload.

You should ensure that no team members are overloaded with too many responsible tasks. Discuss within your team and decide on a reasonable workload together, considering all their daily responsibilities. Should certain team members or roles have too many responsibilities, you must consider increasing your team or re-assign responsibilities.

For example, you may have a product manager who needs to keep an eye on the market, drive product-market fit , perform discovery activities, and support sales and marketing while also acting as the product owner for the development team.

This will lead the team to increased frustration in the development team’s lack of availability of the product manager, so it might be a good idea to consider bringing a dedicated product owner on board. This way, they will dedicate their full attention to the team, while the product manager can focus on the more strategic aspects of the product.

RACI template

Lastly, here is a template that you can use to define your product team’s own RACI chart. Make a copy of the Google Sheet, add your tasks, and define your roles and responsibilities.

While the RACI chart is a tool originating from traditional project management practices, it is as valuable for product management. It is lightweight, but holds a lot of relevant information. It also increases transparency around roles and responsibilities and eases the onboarding process of new team members.

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How to Streamline Roles and Promote Team Collaboration with an Effective RACI Chart

Last Updated November 7, 2019

Imagine for a moment a homeowner’s association is tasked with a project of throwing a summer barbeque bash for the neighborhood. Everyone shows up at the same time, mills around trying to find a task and chaos ensues as homeowners arrive before food has been cooked. Now imagine that same team of people tasked with the same project, but in this team, each individual is assigned a specific task, from grilling to game planning. Now, this team can immediately get to work, and it’s much more likely the neighborhood bash will be an enjoyable event.

In any project, whether the task at hand is throwing a summer party or helping an organization with a technology transformation, when roles and responsibilities for everyone involved are clear, the project is more likely to be executed smoothly. In addition, project managers are often beholden to multiple stakeholders, and keeping those key players informed and involved throughout the lifecycle of the project is critical to project success.

What is a RACI Chart?

One method for streamlining accountability on a project is the use of a responsibility assignment matrix, specifically, a RACI chart. The RACI chart describes how the matrix assigns each task or deliverable, assigns an owner, and denotes who else is involved, ultimately classifying involved parties into four categories: responsible, accountable, consulted and informed. This approach is widespread among project managers, according to “ A Guide to the Project Management Body of Knowledge ( PMBOK® Guide), Seventh Edition,” as it can eliminate confusion on who is in charge of, or accountable for, a specific project task.

The first step to developing an effective RACI chart is understanding what the various categories represent:

Responsible: This is the active party, the person who executes the task or completes the deliverable. This includes doing the work as well as making decisions. In some cases, a task or deliverable may have more than one responsible party; for best results, this role should be limited to as few people as possible for each task.

Accountable: This individual is responsible for ensuring the task is completed correctly and meets all specifications. While this person is not in charge of doing the work, they do delegate it and are responsible for approving the job that is completed. Like the previous category, the number of people assigned to this role should be limited for each task to ensure clarity in ownership.

Consulted: This group, typically comprising of subject matter experts, help provide supporting information for the person(s) responsible for working on the task. Communication with this group is two-way and is required before the task can be completed.

Informed: This group of people must be kept updated on the progress of the task or deliverable, as these individuals may be impacted by the outcomes. They typically are not involved in the feedback or review cycle but should be contacted after a decision or action is made.  

How RACI Charts Influence Project Outcomes

Planning is a critical component of project management and project success, and establishing roles and responsibilities is a key component of the planning stage. Implementing a RACI chart offers several advantages for project participants, including:

• Streamlined communication – All parties know precisely who to ask questions to, who to consult and who to inform.
• A controlled set of stakeholders – By pre-defining these responsibilities, project managers are less likely to be left juggling an unnecessary amount of opinions. Distinguishing consulters from informers can also help avoid a bulk amount of stakeholder feedback .
• Manages fatigue and overwhelm – Although project managers may hold many responsibilities, a RACI chart helps outline other owners, knowledgeable parties and accountable players to help lift some of the burden and ensure everyone stays informed.
• Establishes expectations up front – Creating a RACI chart can also help manage conflict later in the project lifecycle, as everyone should visually understand their roles and responsibilities on a project from the beginning.

Using a RACI Chart

RACI charts don’t have to be complex; they can be as simple as listing the work to be done, the resources assigned and the responsibility they hold. Follow these steps to create your own RACI chart:

Step 1: List the names of the people involved in the project – You’ll need to determine if roles or specific names are appropriate. For example, if a single person holds multiple roles, you could specify by role, whereas if multiple people hold similar titles, you might need to specify by name.

Step 2: Break down specific tasks or deliverables – Although this should be a breakdown of the project, it’s important to balance this to ensure the chart doesn’t become cumbersome or impossible to manage.

Step 3: Assign a role to teach task or deliverable, using RACI – Each task must have someone assigned to be responsible and someone assigned to be accountable.

Step 4: Seek buy in from the team – Gather your team, ensure everyone agrees with their assigned roles and responsibilities and allow for questions.

Step 5: Communicate with project stakeholders – Once you have team buy in, meet with stakeholders and get their buy in to establish expectations up front and avoid conflict down the road. The process of collecting buy in from the team and stakeholders is critical to achieving an effective project.

Step 6: Refer to the RACI – This step may seem obvious, but it doesn’t do any good to plan for the project and then not follow the chart as intended. Ensure everyone continues to adhere to the roles outlined in the RACI chart – remember, they approved them in the planning stage.

RACI Chart Example

Let’s go back to that neighborhood (NBHD) barbecue party and create a RACI chart for the homeowner association (HOA).

Keep in mind that RACI charts are useful in many cases but may not be needed in every case.  For example, rapid projects with small teams likely don’t need introduced complexity. Some Agile projects (depending on scope and timeline) may also have an implied role matrix as some roles are pre-defined, such as the Scrum team .

In implementing a RACI chart, a project manager should immediately set expectations for everyone involved. This includes ensuring team members understand when and what they must do, experts knowing when their opinion will be solicited, and stakeholders understanding where they will be informed, and where they will be consulted for input. Following the method can help establish a foundation for a streamlined project and pave the way to stronger relationships that can make a meaningful difference in the success of the project.

PMBOK is a registered mark of the Project Management Institute, Inc.

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Creating a RACI matrix: define your team’s roles and responsibilities

How to create and use a RACI Matrix

Working on a complex project can be rewarding, stimulating, and, if we’re honest, frustrating. There are numerous elements that could grind a project manager’s gears, from delays to cost overruns . But there is little more frustrating than realizing too late that a member of the team had no idea they were accountable for a key component of the project.

No matter how talented and experienced your people are or how in-depth your project plan is, mistakes will happen if your team aren’t clear about their responsibilities. Being transparent about who needs to be involved and to what extent on every project task may feel overwhelming. Luckily, there’s a simple way to lay out this information and avoid mistakes due to unclear assignments.

Let’s discuss the RACI matrix and how you can create one to define your team’s roles and responsibilities.

What is a RACI matrix?

A RACI matrix, or RACI chart , is a type of responsibility assignment matrix that’s about making responsibilities clear. In simple terms, it’s a diagram that lists the tasks that make up a project and who is responsible for each task.

RACI stands for the four different levels of involvement someone may have in a task, Responsible, Accountable, Consulted, and Informed. The ‘tasks’ listed can cover anything from a major milestone to a decision.

The function of a RACI matrix is to limit confusion about who should be responsible for what. That means less finger-pointing, avoidance tactics, and miscommunication when a difficult call needs to be made or an error needs addressing.

Is using a RACI matrix right for my project?

Using a RACI matrix will benefit most projects. However, there are specific scenarios where it’ll work best, such as for projects with a huge number of moving parts. If your project requires the team to work on multiple larger tasks at the same time or involves more than one external team, a RACI matrix will help keep you keep on top of everything.

Equally, that means there are certain projects where a RACI matrix may not be suitable. For example, if a project is relatively simple and responsibilities can be communicated clearly through the team using traditional methods, setting up a RACI matrix might not be worth it. Weigh up the effort of creating a RACI matrix against the benefits it could offer your team and project to see if it’s worthwhile.

RACI matrixes are popular in project management but can also be used for software development, business analytics, construction, and website development, among other disciplines.

A RACI matrix can be useful if:

• The team are known to forget their responsibilities
• There is a culture of finger-pointing
• Workload ends up being unevenly distributed throughout the team
• There is high turnover in the team
• Approvals need to be made quickly and efficiently

A RACI matrix may not be helpful if:

• The project is small
• The same team members retain the same level of responsibility across all tasks
• The team is known for its effective communication and healthy approach to collaboration

How to create a RACI matrix for project management

Now that we know what a RACI matrix is and what it is helpful for (and not helpful for) let’s delve into how to create one. This will help you better manage your team as a project manager.

Understand the definitions

The first step is to understand what RACI means. As mentioned, this stands for Responsible, Accountable, Consulted, and Informed. Let’s go over what each means.

• Responsible : This team member is responsible for doing the work needed to complete the task. This assignment is task-oriented and can relate to a whole team. There has to be at least one responsible person assigned to each task.
• Accountable : This team member takes charge of delegating work and has the final say over whether a task has been completed successfully or not. This assignment is focused on the outcome. There should only be one Accountable person for each task.
• Consulted : There may be multiple Consulted individuals assigned to a task, depending on its complexity. These people offer unique insights based on expertise and provide feedback on the task based on this and how it’ll impact their future work.
• Informed : Informed team members need to be kept up-to-date on progress relating to the specific task, but they aren’t directly involved. They may be a client or executive.

As you may have realized, some assignments will overlap. For example, the person Responsible for completing the task may also be Accountable.

List the deliverables

Now, let's go over how to set your RACI chart up. The easiest way to do this is using an Excel spreadsheet or similar. 

Start by writing ‘Task’ in cell A1. Then, input the list of tasks that need to be completed in your project below the title in column A. This can include individual tasks, deliverables, milestones, and decisions.

If you like to organize your tasks by category or project phase, you can add headers for each section to divide these up. How you approach this will be dependent on your style of organization and existing project plan.

Add your team members

Starting from cell B1, list the project roles that make up your team in the cells along row 1. You can include your team members’ names if this is going to be helpful. You may also need to include external stakeholders or suppliers if they will be involved, including those who will be assigned as Informed.

Assign the RACI values

The final step is to assign a RACI value to each person against each task. Go through the sheet line by line and populate your matrix based on the definitions above. 

You can even create a separate tab that details the definitions of each role. This can then be used to create a simple drop-down using ‘ data validation ’ if you find this easier than typing in each letter. It can also help avoid errors or typos.

Here’s an example of what your RACI matrix could look like:

Now that you can see it laid out, it becomes clear why a RACI matrix is helpful in keeping track of responsibilities and tasks.

Inform your team

RACI matrixes work best when they’re created collaboratively. Instead of simply assigning responsibilities and expecting your team to fall into line, inform them of your decisions and give them an opportunity to give feedback before finalizing your chart.

Remember the rules and best practices

When you begin using a new, unfamiliar tool, it's best to start out by abiding by its rules and best practices.

Best practices:

• Make sure everyone has been made aware of the decisions you’ve made about their responsibilities before the RACI matrix is approved.
• Don’t add meetings, reports, or repetitive to-dos to the matrix; instead, focus on tasks, milestones, and important decisions that drive the project forward.
• Make sure the task names are recognizable when compared with your project plan. This will help you track progress and pull up details quickly.

RACI rules:

• Don’t forget to assign at least one Responsible person to each task
• Don’t be tempted to assign more than one Accountable person. Doing so will confuse matters further!
• Dole out Responsible assignments carefully so no one person is overwhelmed by tasks.
• Involve every member of your team in every task, even if that means a few individuals are informed on multiple tasks.

The benefits of using a RACI matrix

If you’re not already convinced of the merits of creating a RACI matrix for your current project, let’s review some of the top benefits.

Clarified roles

Clearly delegating roles is fundamental to running a successful project. Even the tightest-knit teams can have slips in communication, and even the most organized individuals can forget a task on their to-do list. 

A RACI matrix makes it clear in no uncertain terms what needs to be done and the role that each person plays in each of those tasks. WIth inflexible definitions supplied for the four assigned roles, there is little room for misinterpretation.

Smooth handovers

When a key team member leaves and a new starter needs to be onboarded, mistakes can occur. With a RACI chart in place, it’s clear what responsibilities a leaver’s replacement needs to be picked up — including what is urgent — even if they’ve forgotten to add it to their handover. A RACI chart can even come in handy when workload needs to be reassigned during paid time off.

Ensures all eventualities are covered

As a project commences, it can be easy to get caught in the weeds. By setting your RACI chart up early, you protect yourself from tasks or decisions being forgotten along the way.

Optimized communication

Having a RACI chart available for all team members to refer to makes it clear when stakeholders need to be informed of updates so they aren’t left out of the loop unintentionally.

Alternatives to RACI charts

There are several alternative ways to approach responsibility assignments. If a RACI chart isn’t the right fit for you, you may be interested in trying out one of these alternatives.

• DACI chart: Standing for Driver, Approver, Contributor, and Informed, a DACI chart seeks to make decision-making roles clear.
• RAPID: Standing for Recommend, Agree, Perform, Input, Decide, a RAPID responsibility matrix is useful in clarifying authority versus accountability, with the Recommend role taking charge of driving the project forward and the Decide role having ultimate responsibility for decision-making.
• RASCI matrix: This final alternative adds a fifth role to the RACI mix. The Supportive role isn’t responsible for any outcomes, instead acting as a support to the Responsible individual.

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The Defense Acquisition Encyclopedia

Program Management

A Responsibility Assignment Matrix (RAM) describes the participation of various organizations, people, and their roles in completing tasks or deliverables for a project. The Program Manager (PM) uses it to clarify roles and responsibilities in a cross-functional team , projects, and processes. A RAM has four primary assignments: Responsible , Accountable , Consulted , and Informed (also called a RACI matrix).

Definition: A Responsibility Assignment Matrix (RAM) describes the role and responsibilities of various people and/or organizations in completing specific tasks for a project.

Responsible, Accountable, Consulted, and Informed (RACI) Matrix

A RAM is called a Responsible, Accountable, Consulted, and Informed (RACI) matrix. The PMBOK Guide 4th Edition defines RACI as a RAM that illustrates the connections between work packages or activities and project team members. In fundamental terms, RAM refers to the framework in place to distribute duties to individuals where, in a RACI, each team member is assigned a role based on one of the four roles. On larger projects, RAMs can be developed at various levels.

• Responsible (R): Those who do the work to achieve the task. There is typically one role with a participation type of responsibility, although others can be delegated to assist in the work required.
• Accountable (A): The one ultimately accountable for the correct and thorough completion of the deliverable or task, and the one to whom Responsible is accountable. In other words, an Accountable must sign off (Approve) on work that Responsible provides. There must be only one Accountable specified for each task or deliverable.
• Consulted (C): Those whose opinions are sought and with whom there is two-way communication.
• Informed (I): Those who are kept up-to-date on progress, often only on completion of the task or deliverable, and with whom there is just one-way communication.

Benefit of Utilizing a Responsibility Assignment Matrix (RAM)

The RAM holds substantial advantages for project managers by clarifying the importance of their processes within the team. It fosters a sense of collective contribution among all employees, eliminating the sense of isolation. This project management technique, the RAM, empowers every team member to grasp the broader context of their work. Instead of simply instructing an administrative assistant to collect phone numbers without context, you can refer them to this valuable resource. By using the RAM, employees become more engaged in achieving positive results as they comprehend the alignment of their contributions with the company’s overall operations.

Responsibility Assignment Matrix (RAM) Goal in Project Management

A RAM is used in project management as a communication tool to ensure that work tasks are designated as a responsible agent. A RAM can define what a project team is responsible for within each component of the Work Breakdown Structure (WBS) . It could also be used within a working group to designate roles, responsibilities, and levels of authority for specific activities. The matrix format shows all activities associated with one person and all people associated with one activity. This ensures that only one person is accountable for any task to avoid confusion.

Responsibility Assignment Matrix (RAM) Tutorial

Responsibility Assignment Matrix (RAM) Standard Format

A RAM is displayed as a chart that illustrates the interaction between work packages that need to be done and project team members. Typically, the list of objectives is on the left-hand column with the project team member names across the top. Each work package will be assigned to the appropriate project team member. The chart aids in communication among the project team members.

No one should typically have more than one degree of responsibility for any given deliverable or activity group in the RAM chart. To simplify things, we’ve assigned each participant in this scenario a certain amount of commitment. However, there is frequently white space when you create a genuine model for more than four people. In some situations, it’s okay to have someone with secondary responsibility but not primary.

Responsibility Assignment Matrix (RAM) Template

Template: responsibility assignment matrix (ram) (excel), 6 steps to developing a responsibility assignment matrix (ram).

Below is a list of the 6 (six) most common steps in developing a Responsibility Assignment Matrix (RAM).

• Step 1: List all project tasks and deliverables
• Step 2: Identify all project stakeholders
• Step 3: Determine the responsibility and accountability level for each task and deliverable
• Step 4: Assign stakeholders to each task
• Step 5: Assign overall stakeholder
• Step 6: Ensure all stakeholder know their responsibility

Developing Responsibility Assignment Matrix (RAM) Matrix Best Practices

Below is a list of best practice topics that can help Program Managers effectively build and use a Responsibility Assignment Matrix.

• One stakeholder is in charge per task.
• The least amount of people accountable, the better.
• Be Efficient with Meetings.
• Constant Communication.
• Stakeholders agree on final RAM

Responsibility Assignment Matrix (RAM) Lessons Learned

A Responsibility Assignment Matrix (RAM) is a tool used in project management to identify and clarify the roles and responsibilities of the different people or groups working on a project. The goal of making a RAM is to make sure that all tasks are done and that responsibilities don’t overlap or get missed. Here are some things you can learn to make sure your RAM is built right:

• Define the project’s goals and scope in detail:  Before making a RAM, it’s important to have a clear idea of the project’s goals and scope. This will help make sure that all necessary tasks are included and that the responsibilities are in line with the overall project goals.
• Find out who all the stakeholders are and what their roles are:  A RAM should have a list of all the people or groups involved in the project, such as internal team members, external partners, and customers. There should be roles and responsibilities for each stakeholder.
• Give each stakeholder specific tasks and responsibilities:  Instead of giving each stakeholder a general role, it is important to give them specific tasks and responsibilities. This will help make sure that no one’s responsibilities get mixed up or left out.
• Make sure that all stakeholders know about and understand the RAM:  It is important to make sure that all stakeholders know about and understand the RAM. This can be done by having regular meetings and giving updates, as well as by putting the RAM in writing.
• Review and update the RAM often: As the project moves forward, it may be necessary to review and update the RAM. This can help make sure that the RAM stays correct and helps the project reach its goals.

Difference Between a Responsibility Assignment Matrix (RMA) and a Responsible, Accountable, Consulted, and Informed (RACI) Matrix

The PMBOK Guide 4th Edition defines RACI as a RAM that is used to illustrate the connections between work packages in a Work Breakdown Structure (WBS) and project team members. The difference between a RAM matrix and RACI matrix is:

• A Responsibility Assignment Matrix (RAM) describes the participation of various organizations, people, and their roles in completing tasks or deliverables in a Work Break Down Structure (WBS) for a project.
• A Responsible, Accountable, Consulted, and Informed (RACI) matrix is used on projects where multiple groups of people as assigned a task. It helps on larger projects with a lot of people and organizations. It also helps with outside stakeholders and their responsibilities on a project.
• A RACI can have multiple RAM within it.

AcqLinks and References:

• Template: Responsibility Assignment Matrix (RAM) Template (Word)
• Template: Responsibility Assignment Matrix (RAM) Template (Excel)

Updated: 1/11/2024

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Types of Matrices classifies matrices in different categories based on the number of rows and columns present in them, the position of the elements, and also the special properties exhibited by the Matrix. Matrix is a rectangular array of numbers in which elements are arranged in rows and columns. Each element is identified as a ij where i and j indicate the row and column number respectively for the element. We have different types of matrices which are classified on the basis of the number of rows and columns, the elements present in them, the order of the matrix, and the properties shown by the matrix. In this article, we will learn about different types of matrices in detail along with a brief introduction to matrices.

What is Matrix?

A rectangular array of numbers, symbols, or any type of entries arranged in the form of rows and columns is called Matrix . A matrix has one or more than one number of rows and columns. The horizontal arrangement of elements is called Row and the vertical arrangement is called the column of matrix. Each element of a matrix is identified by its position which is given by the row and column in which it is present. A matrix is represented by [A] m×n , where m is the number of rows and n is the number of columns present in a matrix. Thus an element of the matrix can be represented as a ij where i and j are the i th row and j th column to which an element belongs. Matrix A can be written as:

[Tex]\begin{bmatrix}  a_{11}& a_{12}  & a_{13}  &.  & . &.  &  a_{1n}\\ a_{21} & a_{22}  & a_{23}  & . &.  &.  & a_{2n} \\ a_{31}& a_{32}  & a_{33}  & . &.  &.  & a_{3n} \\  .&.  &.  &.  &.  &.  &. \\ . &.  &.  &.  &.  &.  &. \\ . &.  &.  &.  &.  &.  &. \\ . & . &.  &.  &.  &.  &. \\  a_{m1}& a_{m2}  & a_{m3}  &.  &.  &.  & a_{mn} \end{bmatrix}_{m\times n}  [/Tex]

Example of a Matrix

An example of a Matrix is mentioned below:

[Tex]\begin{bmatrix}  1& 5  &8   &5 \\ 3& 4  &0   &  12 \\ 7& 2  &3   & 10 \\ \end{bmatrix}_{3\times 4}  [/Tex]

Here we can see that we have a matrix of order 3 ⨯ 4 which means there are 3 rows in the matrix and 4 columns in the matrix.

Types of M atrix

There are many types of matrices depending on the elements in the matrix, order, and certain sets of conditions. The different types of matrices are mentioned below:

Singleton Matrix

Null matrix, column matrix, horizontal matrix, vertical matrix, rectangular matrix, square matrix, diagonal matrix.

• Scalar Matrix

Identity Matrix

Upper triangular matrix, lower triangular matrix, singular matrix, non singular matrix, symmetric matrix, skew symmetric matrix, orthogonal matrix, idempotent matrix, nilpotent matrix, periodic matrix.

• Involutory Matrix

Hermitian Matrix

Skew hermitian matrix, boolean matrix, stochastic matrix.

Let’s learn the above types of matrices in detail

A matrix that has only one element is called a singleton matrix. In this type of matrix number of columns and the number of rows is equal to 1. A singleton matrix is represented as [a] 1⨯1 .

Example of Singleton Matrix

[Tex]\begin{bmatrix}  5 \end{bmatrix}_{1\times 1}  [/Tex]

In the above example of the Singleton Matrix, there is only one element 5. Hence there is only one column and only one row.

A matrix whose all elements are zero is called a Null Matrix. A null matrix is also called a Zero Matrix because its all elements are zero. An example of a null matrix is mentioned below:

Example of Zero Matrix

[Tex]\begin{bmatrix}  0& 0 &0    \\ 0& 0 &0  \\ 0& 0  &0   \\ \end{bmatrix}_{3\times 3}  [/Tex]

In the above example of a null or zero matrix, all the elements are zero. Hence the given example is a matrix of order 3 ⨯ 3 whose all elements are zero.

A matrix that contains only one row and any number of columns is known as a row matrix. A row matrix is represented as [a]1⨯n where 1 is the number of row and n is the number of columns present in a row matrix. An example of a row matrix is given below

Example of Row Matrix

[Tex]\begin{bmatrix}  1& 3&7 \end{bmatrix}_{1\times 3}  [/Tex]

In the above example of a row matrix, the number of rows is 1, and the number of columns is 3. Hence the order of the matrix is 1 ⨯ 3.

A matrix that contains only one column and any number of rows is called a Column Matrix. A Column Matrix is represented as [a]n⨯1 where n is the number of rows and 1 is the number of columns. An example of a column matrix is given below:

Example of Column Matrix

[Tex]\begin{bmatrix}  1\\   15 \\ 4\\ 5 \\ \end{bmatrix}_{4\times 1}  [/Tex]

In the above example of a column matrix the number of rows is 4 and the number of columns is 1 thus making it a matrix of order 4 ⨯ 1.

A matrix in which the number of rows is lower than the number of columns is called a Horizontal Matrix. columns

Example of Horizontal Matrix

[Tex]\begin{bmatrix} 1 & 2 & 3 & 4\\ 5& 6& 7& 8 \end{bmatrix}_{2\times4} [/Tex]

In the above matrix, the number of rows is 2 while the number of columns is 4 thus making it a horizontal matrix.

The matrix in which the number of rows exceeds the number of columns is called a Vertical Matrix. A Vertical matrix is represented as [a]i⨯j where i > j. An example of a Vertical Matrix is mentioned below:

Example of Vertical Matrix

[Tex]\begin{bmatrix} 1 & 2 \\ 3 & 4\\ 5& 6\\ 7& 8 \end{bmatrix}_{4\times2} [/Tex]

In the above matrix, the number of rows is 4 while the number of columns is 2 thus making it a Vertical matrix.

A matrix that does not have an equal number of rows and columns is known as a Rectangular Matrix. A rectangular matrix can be represented as [A] m×n where m ≠ n. An example of a rectangular matrix is mentioned below:

Example of Rectangular Matrix

[Tex]\begin{bmatrix}  1& 3  &7   &15 \\ 3& 4  &6  &  11 \\ 5& 2  &9  & 8 \\ \end{bmatrix}_{3\times 4}  [/Tex]

In the above example, we see that the number of rows is 3 while the number of columns is 4 i.e. both are unequal thus making it a rectangular matrix. We can say that both horizontal and vertical matrices are examples of rectangular matrices.

A matrix that has an equal number of rows and an equal number of columns is called a Square Matrix. Generally, the representation used for the square matrix is [A] n×n . An example of Square Matrix is mentioned below:

Example of Square Matrix

[Tex]\begin{bmatrix}  8& 3  &2    \\ 6& 4  &6   \\ 5& 7  &9   \\ \end{bmatrix}_{3\times 3}  [/Tex]

In the above example of Square Matrix, both the number of rows and columns are 3, thus making them seem like a square structure.

A matrix that has all elements as 0 except diagonal elements is known as a diagonal matrix. A Diagonal Matrix is only possible in the case of a Square Matrix. An example of a Diagonal Matrix is mentioned below:

Example of Diagonal Matrix

[Tex]\begin{bmatrix}  8& 0 &0    \\ 0& 4  &0  \\ 0& 0  &9   \\ \end{bmatrix}_{3\times 3}  [/Tex]

In the above example, the diagonal elements are 8, 4, and 9 and the rest elements are zero.

Scalar Matrix 

A diagonal matrix whose all diagonal elements are non-zero and the same is called a Scalar Matrix. Scalar Matrix is a kind of diagonal matrix where all diagonal elements are the same. Identity Matrix is a special case of Scalar Matrix.

Example of Scalar Matrix

[Tex]\begin{bmatrix}  4& 0 &0    \\ 0& 4  &0  \\ 0& 0  &4   \\ \end{bmatrix}_{3\times 3}  [/Tex]

In the above example, the given matrix is a diagonal matrix whose all diagonal elements are 4, and hence, this is an example of a Scalar Matrix.

A diagonal matrix where all the diagonal elements are 1 and all non-diagonal elements are 0 is called an Identity Matrix. The Identity Matrix is called Unit Matrix. The identity matrix or unit matrix always has an equal number of rows and columns.

Example of Identity Matrix

[Tex]\begin{bmatrix}  1& 0 &0    \\ 0& 1 &0  \\ 0& 0  &1   \\ \end{bmatrix}_{3\times 3}  [/Tex]

In the above diagonal matrix of order 3 ⨯ 3, all the diagonal elements are 1, and non-diagonal elements are zero. Hence this diagonal matrix is an Identity Matrix.

Triangular Matrix

A square matrix in which the non-zero elements form a triangular below and above the diagonal is called a Triangular Matrix. Based on the triangle formed below or above the diagonal, the triangular matrix is classified as:

Let’s learn them in detail.

A square matrix in which all the elements below the diagonal are zero and the elements from the diagonal and above are non-zero elements is called an Upper Triangular Matrix. In an Upper Triangular Matrix, the non-zero elements form a triangular-like shape.

Example of Upper Triangular Matrix

[Tex]\begin{bmatrix}  8& 5 &6    \\ 0& 4  &7  \\ 0& 0  &9   \\ \end{bmatrix}_{3\times 3}  [/Tex]

In the above example of the Upper Triangular Matrix, all the elements below the diagonal are zero.

A square matrix in which all the elements above the diagonal are zero and the elements from the diagonal and below are non-zero elements is called a Lower Triangular Matrix. In a Lower Triangular Matrix, the non-zero elements form a triangular-like shape from the diagonal and below.

Example of Lower Triangular Matrix

[Tex]\begin{bmatrix}  8& 0  &0    \\ 6& 4  &0  \\ 5& 7  &9   \\ \end{bmatrix}_{3\times 3}  [/Tex]

In the above example of the lower triangular matrix, all the elements above the diagonal are zero.

A singular matrix is referred to as a square matrix whose determinant is zero and is not invertible . If det A = 0, a square matrix “A” is said to be singular; otherwise, it is said to be non-singular.

Example of Singular Matrix

[Tex]A= \left[\begin{array}{ccc} 3 & 6 & 9\\ 6 & 12 & 18\\ 2 & 4 & 6 \end{array}\right] [/Tex]

⇒ |A| = 3(12×6 – 18×4)-6(6×6 – 18×2) + 9(6×4 – 12×2)

⇒ |A| = 3(72 – 72) – 6(36 – 36) + 9(24 – 24)

⇒ |A| = 3×0 – 6×0 + 9×0 = 0

A Non-Singular matrix is defined as a square matrix whose determinant is not equal to zero and is invertible.

Example of a Non-Singular Matrix

[Tex]|A| = \left[\begin{array}{cc} 1 & 5\\ 9 & 8 \end{array}\right] [/Tex]

⇒ |A| = 8×1 – 9×5 = 8 – 45 = -37

A square matrix “A” of any order is defined as a symmetric matrix if the transpose of the matrix is equal to the original matrix itself, i.e., A T = A.

Example of Symmetric Matrix

[Tex]\left[\begin{array}{cc} 2 & 1\\ 1 & 2 \end{array}\right] [/Tex]

A square matrix “A” of any order is defined as a skew-symmetric matrix if the transpose of the matrix is equal to the negative of the original matrix itself, i.e., A T = -A.

Example of Skew Symmetric Matrix

[Tex]\left[\begin{array}{ccc} 0 & 3 & 5\\ -3 & 0 & -2\\ -5 & 2 & 0 \end{array}\right] [/Tex]

A square matrix whose transpose is equal to its inverse is called Orthogonal Matrix. In an Orthogonal Matrix if A T = A -1 then AA T = I where I is the Identity Matrix.

Example of Orthogonal Matrix

[Tex]A = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} [/Tex]

and  [Tex]A^T = \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix} [/Tex]

[Tex]\Rightarrow A \times A^T = \begin{bmatrix} \cos^2(\theta)+\sin^2(\theta) & \cos(\theta)\sin(\theta)-\cos(\theta)\sin(\theta) \\ \sin(\theta)\cos(\theta)-\cos(\theta)\sin(\theta)& \cos^2(\theta)+\sin^2(\theta)\end{bmatrix} [/Tex]

[Tex]\Rightarrow A \times A^T = \begin{bmatrix}1&0\\0&1 \end{bmatrix} = I_{(2\times 2)} [/Tex]

An idempotent matrix is a special type of square matrix that remains unchanged when multiplied by itself, i.e., A 2 = A.

Example of Idempotent Matrix

[Tex]A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} [/Tex]

Hence,  [Tex]A \cdot A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = A [/Tex]

A Nilpotent is a square matrix that when raised to some positive power results in zero matrix. The least power let’s say ‘p’ for which the matrix yields zero matrix, then it is called the Nilpotent Matrix of power ‘p’.

Example of Nilpotent Matrix

[Tex]A = \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix} [/Tex]

[Tex]\Rightarrow A^2= \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix}\cdot \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix} [/Tex]

[Tex]\Rightarrow A^2 = \begin{bmatrix} 0 & 0 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} [/Tex]

and  [Tex]A^3 = A \cdot A^2  [/Tex]

[Tex]\Rightarrow A^3 = \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} [/Tex]

Hence, A is a Nilpotent Matrix of index 3.

A periodic matrix is a square matrix which exhibits periodicity, i.e. when raised to some power let’s say p+1 then A p+1 = A. If p = 1 then A 2 = A it means A is an Idempotent Matrix. Thus we can say that Idempotent Matrix is a case of Periodic Matrix.

Example of Periodic Matrix

The above square matrix is a Periodic Matrix of Period 2, where p = 1.

Involuntary Matrix

An involuntary matrix is a special type of square matrix whose inverse is the original matrix itself, i.e., P = P -1 , or, in other words, its square is equal to an identity matrix i.e. P 2 = I.

Example of Involuntary Matrix

[Tex]A = \begin{bmatrix} 2 & 1 \\ -3 & -2 \end{bmatrix} [/Tex]

A complex square matrix is called a Hermitian Matrix if the conjugate transpose of the matrix is equal to the original matrix. In this type of matrix the diagonal elements must be a real number.

Example of Hermitian Matrix

[Tex]A = \begin{bmatrix} 3 & 1 – i \\ 1 + i & 4 \end{bmatrix} [/Tex]

A complex square matrix is called a Skew Hermitian Matrix if the conjugate transpose of the matrix is equal to the negative of the original matrix. In this type of matrix the diagonal elements can be either 0 or a complex number but can not be real number other than 0.

Example of Skew Hermitian Matrix

[Tex]A = \begin{bmatrix} 2i & 2i & -3i \\ -2i & 0 & 4 \\ 3i & -4 & 0 \end{bmatrix} [/Tex]

The matrix which represents the binary relationship and takes 0 and 1 as its element is called Boolean Matrix.

Example of Boolean Matrix

[Tex]\begin{bmatrix}1 & 0 &1\\0 & 1 & 0\\1&1&0 \end{bmatrix} [/Tex]

A Square Matrix which represents the probability data i.e. a non-negative element such that the summation of each row is 1.

Example of Stochastic Matrix

[Tex]\begin{bmatrix}0.2 & 0.5 &0.3\\0.1 & 0.3 & 0.6\\0.4&0.2&0.4 \end{bmatrix} [/Tex]

Trace of a Matrix

The sum of diagonal elements of a matrix is known as the trace of a matrix. The Trace of a matrix A can be represented as tr(A). The Trace of a matrix can be calculated for a square matrix only.

[Tex]A= \begin{bmatrix}  15& 12 &9 \\ 4& 6 &11  \\ 5& 9  &0   \\ \end{bmatrix}_{3\times 3}   [/Tex] tr(A) = 15 + 6 + 0 = 21

Properties of a Trace of Matrix

i) Trace of the sum of two matrices is equal to the sum of a trace of an individual matrix.

Explanation:

Mathematically it can be written as tr(A+B) = tr(A) + tr(B) [Tex]A= \begin{bmatrix}  15& 12 &9 \\ 4& 6 &11  \\ 5& 9  &0   \\ \end{bmatrix}_{3\times 3}   [/Tex] tr(A) = 15 + 6 + 0 = 21  [Tex]B= \begin{bmatrix}  4& 3 &7 \\ 8& 1 &2  \\ 5& 6  &1   \\ \end{bmatrix}_{3\times 3}   [/Tex] tr(B) = 4 + 1 + 1 = 6 Now, tr(A)+tr(B) = 21+6 = 27 [Tex]A+B= \begin{bmatrix}  19& 15&16 \\ 12& 7 &13 \\ 10& 15  &1   \\ \end{bmatrix}_{3\times 3}   [/Tex] tr(A + B) = 19 + 7 + 1 = 27 You can see, tr(A) + tr(B) = tr(A + tr(B) Similarly, tr(A – B) = tr(A) – tr(B)

ii) Trace of a matrix that is multiplied by some scalar is equal to the multiplication of the trace of the matrix and scalar.

Mathematically it can be represented as tr(kA) = k tr(A) [Tex]A=2\times \begin{bmatrix}  1& 4 &3    \\ 5& 7 &2  \\ 1& 3  &8   \\ \end{bmatrix}_{3\times 3}   [/Tex] tr(2 × A) = 2 + 14 + 16 = 32 [Tex]2\times A= \begin{bmatrix}  2& 8 &6    \\ 10& 14 &4  \\ 2& 6  &16   \\ \end{bmatrix}_{3\times 3}   [/Tex] 2 × tr(A) = 2 * (1 + 7 + 8)                 = 32 You can see tr(2 × A) = 2 × tr(A)

Read More, ,

Matrix Multiplication Adjoint of a Matrix Minors and Cofactors

FAQs on Types of Matrices

Q1: what are matrices and their types.

Matrices are arrangements of elements or numbers or symbols in the form of an array. The different types of matrices are: Row Matrix Column Matrix Horizontal Matrix Vertical Matrix Singleton Matrix Null Matrix Rectangular Matrix Square Matrix Diagonal Matrix Identity Matrix Scalar Matrix Triangular Matrix Upper Triangular Matrix Lower Triangular Matrix Singular Matrix Non Singular Matrix Symmetric Matrix Skew Symmetric Matrix Orthogonal Matrix Idempotent Matrix Nilpotent Matrix Periodic Matrix Involuntary Matrix Hermitian Matrix Skew Hermitian Matrix Boolean Matrix Stochastic Matrix

Q2: What is a Square Matrix?

A Square matrix is a matrix in which the number of rows and columns are equal.

Q3: What is a Diagonal Matrix?

A square matrix in which all non-diagonal elements are zero is called a Diagonal Matrix.

Q4: What is Unit Matrix?

A diagonal matrix in which all the diagonal elements are 1 is called a Unit Matrix. A unit matrix is also called an Identity Matrix.

Q5: What is a Null Matrix?

A matrix whose all elements are zero is called a Null Matrix. A null matrix is also called a zero matrix.

Q6: What is a Symmetric and Skew Symmetric Matrix?

A matrix whose transpose is equal to the matrix itself is called a Symmetric Matrix while a matrix whose transpose is equal to the negative of the matrix is called a Skew Symmetric Matrix.

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7.6: Matrices and Matrix Operations

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Learning Objectives

• Find the sum and difference of two matrices.
• Find scalar multiples of a matrix.
• Find the product of two matrices.

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. Table \(\PageIndex{1}\) shows the needs of both teams.

A goal costs \($300\); a ball costs \($10\); and a jersey costs \($30\). How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

Finding the Sum and Difference of Two Matrices

To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named \(A\), \(B\), and \(C\) are shown below.

\[ \begin{align*} A&=\begin{bmatrix} 1& 2 \\ 3 & 4 \\ \end{bmatrix} \\[4pt] B &=\begin{bmatrix} 1 & 2 & 7 \\ 0 & -5 & 6 \\ 7 & 8 & 2 \end{bmatrix} \\[4pt] C &=\begin{bmatrix} -1 & 3 \\ 0 & 2 \\ 3 & 1 \end{bmatrix} \end{align*}\]

A matrix is often referred to by its size or dimensions: \(m×n\) indicating \(m\) rows and \(n\) columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix \(A\) identified as \(a_{ij}\),we look for the entry in row \(i\),column \(j\). In matrix \(A\), shown below, the entry in row \(2\), column \(3\) is \(a_{23}\).

\[A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33} \end{bmatrix} \nonumber\]

• A square matrix is a matrix with dimensions \(n × n\), meaning that it has the same number of rows as columns. The \(3×3\) matrix above is an example of a square matrix.
• A row matrix is a matrix consisting of one row with dimensions \(1 × n\). \[\begin{bmatrix} a_{11} & a_{12} & a_{13} \end{bmatrix} \nonumber\]
• A column matrix is a matrix consisting of one column with dimensions \(m × 1\). \[\begin{bmatrix} a_{11} \\ a_{21} \\a_{31} \end{bmatrix} \nonumber\]

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations .

Definition: MATRICES

A matrix is a rectangular array of numbers that is usually named by a capital letter: \(A\), \(B\), \(C\),and so on. Each entry in a matrix is referred to as \(a_{ij}\),such that \(i\) represents the row and \(j\) represents the column. Matrices are often referred to by their dimensions: \(m × n\) indicating \(m\) rows and \(n\) columns.

Example \(\PageIndex{1}\): Finding the Dimensions of the Given Matrix and Locating Entries

Given matrix \(A\):

• What are the dimensions of matrix \(A\)?
• What are the entries at \(a_{31}\) and \(a_{22}\)?

\[A=\begin{bmatrix} 2 & 1 & 0\\2 & 4 & 7\\3 & 1 & −2 \end{bmatrix} \nonumber\]

• The dimensions are \(3 \times 3\) because there are three rows and three columns.
• Entry \(a_{31}\) is the number at row 3, column 1, which is \(3\). The entry \(a_{22}\) is the number at row 2, column 2, which is \(4\). Remember, the row comes first, then the column.

Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries. To do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions . We can add or subtract a \(3 \times 3\) matrix and another \(3 \times 3\) matrix, but we cannot add or subtract a \(2 \times 3\) matrix and a \(3 \times 3\) matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

ADDING AND SUBTRACTING MATRICES

Given matrices \(A\) and \(B\) of like dimensions, addition and subtraction of \(A\) and \(B\) will produce matrix \(C\) or matrix \(D\) of the same dimension.

such that \(a_{ij}+b_{ij}=c_{ij}\)

\[A−B=D\]

such that \(a_{ij}−b_{ij}=d_{ij}\)

Matrix addition is commutative .

\[A+B=B+A\]

It is also associative .

\[(A+B)+C=A+(B+C)\]

Example \(\PageIndex{2A}\): Finding the Sum of Matrices

Find the sum of \(A\) and \(B\), given

\[A=\begin{bmatrix}a & b\\c & d \end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}e & f\\g & h\end{bmatrix} \nonumber\]

Add corresponding entries.

\[\begin{align} A+B &=\begin{bmatrix}a & b\\c & d\end{bmatrix}+\begin{bmatrix}e & f\\g & h\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}a+e & b+f\\c+g & d+h \end{bmatrix} \nonumber \end{align} \nonumber\]

Example \(\PageIndex{2B}\): Adding Matrix \(A\) and Matrix \(B\)

Find the sum of \(A\) and \(B\).

\[A=\begin{bmatrix}4 &1\\3 & 2 \end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}5 & 9\\0 & 7\end{bmatrix} \nonumber\]

Add corresponding entries. Add the entry in row 1, column 1, \(a_{11}\), of matrix \(A\) to the entry in row 1, column 1, \(b_{11}\), of \(B\). Continue the pattern until all entries have been added.

\[\begin{align} A+B &=\begin{bmatrix}4&1\\3 &2\end{bmatrix}+\begin{bmatrix}5&9\\0&7\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}4+5&1+9\\3+0&2+7\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}9&10\\3&9\end{bmatrix} \nonumber \end{align} \nonumber\]

Example \(\PageIndex{2C}\): Finding the Difference of Two Matrices

Find the difference of \(A\) and \(B\).

\(A=\begin{bmatrix}−2&3\\0&1\end{bmatrix}\) and \(B=\begin{bmatrix}8&1\\5&4\end{bmatrix}\)

We subtract the corresponding entries of each matrix.

\[\begin{align} A−B &=\begin{bmatrix}−2&3\\0&1\end{bmatrix}−\begin{bmatrix}8&1\\5&4\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}−2−8&3−1\\0−5&1−4\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}−10&2\\−5&−3\end{bmatrix} \nonumber \end{align} \nonumber\]

Example \(\PageIndex{2D}\): Finding the Sum and Difference of Two 3 x 3 Matrices

Given \(A\) and \(B\):

• Find the sum.
• Find the difference.

\[A=\begin{bmatrix}2&−10&−2\\14&12&10\\4&−2&2\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}6&10&−2\\0&−12&−4\\−5&2&−2\end{bmatrix} \nonumber\]

• Add the corresponding entries.

\[\begin{align} A+B & =\begin{bmatrix} 2& −10& −2\\14 & 12 & 10\\4 & −2 & 2\end{bmatrix}+\begin{bmatrix}6 & 10 & −2\\0 & −12 & −4\\−5 & 2 & −2\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}2+6 & −10+10 & −2−2\\14+0 & 12−12 & 10−4\\4−5 & −2+2 & 2−2\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix} 8 & 0 & −4\\14 & 0 & 6\\−1 & 0 & 0\end{bmatrix} \nonumber \end{align} \nonumber\]

• Subtract the corresponding entries.

\[\begin{align} A−B &=\begin{bmatrix}2&−10&−2\\14&12&10\\4&−2&2\end{bmatrix}−\begin{bmatrix}6&10&−2\\0&−12&−4\\−5&2&−2\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}2−6 & −10−10 & −2+2\\14−0 & 12+12 & 10+4\\4+5 & −2−2 & 2+2\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−4 & −20 & 0\\14 & 24 & 14\\9 & −4 & 4\end{bmatrix} \nonumber \end{align} \nonumber\]

Exercise \(\PageIndex{1}\)

Add matrix \(A\) and matrix \(B\).

\[A=\begin{bmatrix}2&6\\1&0\\1&−3\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}3&−2\\1&5\\−4&3\end{bmatrix} \nonumber\]

\[\begin{align} A+B&=\begin{bmatrix}2&6\\ 1 &0\\1&−3\end{bmatrix}+\begin{bmatrix} 3&-2 \\1&5 \\-4&3\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}2+3&6+(−2)\\1+1&0+5\\1+(-4)&−3+3\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}5&4\\2&5\\-3&0\end{bmatrix} \nonumber \end{align} \nonumber\]

Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that \(15%\) more equipment is needed in both labs. The school’s current inventory is displayed in Table \(\PageIndex{2}\).

Converting the data to a matrix, we have

\[C_{2013}=\begin{bmatrix}15 & 27\\16&34\\16&34\end{bmatrix} \nonumber\]

To calculate how much computer equipment will be needed, we multiply all entries in matrix \(C\) by \(0.15\).

\[(0.15)C_{2013}=\begin{bmatrix}(0.15)15&(0.15)27\\(0.15)16&(0.15)34\\(0.15)16 &(0.15)34\end{bmatrix}=\begin{bmatrix}2.25 &4.05\\2.4&5.1\\2.4&5.1\end{bmatrix} \nonumber\]

We must round up to the next integer, so the amount of new equipment needed is

\[\begin{bmatrix}3&5\\3&6\\3&6\end{bmatrix} \nonumber\]

Adding the two matrices as shown below, we see the new inventory amounts.

\[\begin{bmatrix}15&27\\16&34\\16&34\end{bmatrix}+\begin{bmatrix}3&5\\3&6\\3&6\end{bmatrix}=\begin{bmatrix}18&32\\19&40\\19&40\end{bmatrix} \nonumber\]

\[C_{2014}=\begin{bmatrix}18&32\\19&40\\19&40\end{bmatrix} \nonumber\]

Thus, Lab A will have \(18\) computers, \(19\) computer tables, and \(19\) chairs; Lab B will have \(32\) computers, \(40\) computer tables, and \(40\) chairs.

SCALAR MULTIPLICATION

Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given

\[A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix} \nonumber\]

the scalar multiple \(cA\) is

\[cA=c\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix} \nonumber\]

\[=\begin{bmatrix}ca_{11}&ca_{12}\\ca_{21}&ca_{22}\end{bmatrix} \nonumber\]

Scalar multiplication is distributive. For the matrices \(A\), \(B\),and \(C\) with scalars \(a\) and \(b\),

\[a(A+B)=aA+aB\]

\[(a+b)A=aA+bA\]

Example \(\PageIndex{3}\): Multiplying the Matrix by a Scalar

Multiply matrix \(A\) by the scalar \(3\).

\[A=\begin{bmatrix}8&1\\5&4\end{bmatrix} \nonumber\]

Multiply each entry in \(A\) by the scalar \(3\).

\[ \begin{align} 3A&=3\begin{bmatrix}8&1\\5&4\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}3⋅8&3⋅1\\3⋅5&3⋅4\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}24&3\\15&12\end{bmatrix} \nonumber \end{align} \nonumber\]

Exercise \(\PageIndex{2}\)

Given matrix \(B\), find \(−2B\) where

\[B=\begin{bmatrix}4&1\\3&2\end{bmatrix} \nonumber\]

\[−2B=\begin{bmatrix}−8&−2\\−6&−4\end{bmatrix} \nonumber\]

Example \(\PageIndex{4}\): Finding the Sum of Scalar Multiples

Find the sum \(3A+2B\).

\[A=\begin{bmatrix}1&−2&0\\0&−1&2\\4&3&−6\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}−1&2&1\\0&−3&2\\0&1&−4\end{bmatrix} \nonumber\]

First, find \(3A\), then \(2B\).

\[ \begin{align} 3A&=\begin{bmatrix}3⋅1&3(−2)&3⋅0\\3⋅0&3(−1)&3⋅2\\3⋅4&3⋅3&3(−6)\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}3&−6&0\\0&−3&6\\12&9&−18\end{bmatrix}\nonumber \end{align} \nonumber\]

\[ \begin{align} 2B&=\begin{bmatrix}2(−1)&2⋅2&2⋅1\\2⋅0&2(−3)&2⋅2\\2⋅0&2⋅1&2(−4)\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−2&4&2\\0&−6&4\\0&2&−8\end{bmatrix}\nonumber \end{align} \nonumber\]

Now, add \(3A+2B\).

\[ \begin{align} 3A+2B&=\begin{bmatrix}3&−6&0\\0&−3&6\\12&9&−18\end{bmatrix}+\begin{bmatrix}−2&4&2\\0&−6&4\\0&2&−8\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}3−2&−6+4&0+2\\0+0&−3−6&6+4\\12+0&9+2&−18−8\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}1& −2&2\\0&−9&10\\12&11&−26\end{bmatrix} \nonumber \end{align} \nonumber\]

Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If \(A\) is an \(m × r\) matrix and \(B\) is an \(r × n\) matrix, then the product matrix \(AB\) is an \(m × n\) matrix. For example, the product \(AB\) is possible because the number of columns in \(A\) is the same as the number of rows in \(B\). If the inner dimensions do not match, the product is not defined.

We multiply entries of \(A\) with entries of \(B\) according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row \(i\) of \(AB\), we multiply the entries in row \(i\) of \(A\) by column \(j\) in \(B\) and add. For example, given matrices \(A\) and \(B\), where the dimensions of \(A\) are \(2 \times 3\) and the dimensions of \(B\) are \(3 \times 3\),the product of \(AB\) will be a \(2 \times 3\) matrix.

\[A=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix} \nonumber \]

\[B=\begin{bmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{bmatrix} \nonumber\]

Multiply and add as follows to obtain the first entry of the product matrix \(AB\).

\[\begin{bmatrix}a_{11}&a_{12}&a_{13}\end{bmatrix} ⋅\begin{bmatrix}b_{11}\\b_{21}\\b_{31}\end{bmatrix}=a_{11}⋅b_{11}+a_{12}⋅b_{21}+a_{13}⋅b_{31} \nonumber \]

\[\begin{bmatrix}a_{11}&a_{12}&a_{13}\end{bmatrix} ⋅\begin{bmatrix}b_{12}\\b_{22}\\b_{32}\end{bmatrix}=a_{11}⋅b_{12}+a_{12}⋅b_{22}+a_{13}⋅b_{32} \nonumber \]

\[\begin{bmatrix}a_{11}&a_{12}&a_{13}\end{bmatrix} ⋅\begin{bmatrix}b_{13}\\b_{23}\\b_{33}\end{bmatrix}=a_{11}⋅b_{13}+a_{12}⋅b_{23}+a_{13}⋅b_{33} \nonumber \]

We proceed the same way to obtain the second row of \(AB\). In other words, row 2 of \(A\) times column 1 of \(B\); row 2 of \(A\) times column 2 of \(B\); row 2 of \(A\) times column 3 of \(B\). When complete, the product matrix will be

\[AB=\begin{bmatrix}a_{11}⋅b_{11}+a_{12}⋅b_{21}+a_{13}⋅b_{31} &a_{11}⋅b_{12}+a_{12}⋅b_{22}+a_{13}⋅b_{32}&a_{11}⋅b_{13}+a_{12}⋅b_{23}+a_{13}⋅b_{33} \\a_{21}⋅b_{11}+a_{22}⋅b_{21}+a_{23}⋅b_{31}&a_{21}⋅b_{12}+a_{22}⋅b_{22}+a_{23}⋅b_{32}&a_{21}⋅b_{13}+a_{22}⋅b_{23}+a_{23}⋅b_{33}\end{bmatrix} \nonumber\]

PROPERTIES OF MATRIX MULTIPLICATION

For the matrice \(A, B\),and \(C\) the following properties hold.

• Matrix multiplication is associative : \[(AB)C=A(BC).\]
• Matrix multiplication is distributive : \[C(A+B)=CA+CB\] \[(A+B)C=AC+BC.\]

Note that matrix multiplication is not commutative.

Example \(\PageIndex{5A}\): Multiplying Two Matrices

Multiply matrix \(A\) and matrix \(B\).

\[A=\begin{bmatrix}1&2\\3&4\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}5&6\\7&8\end{bmatrix} \nonumber\]

First, we check the dimensions of the matrices. Matrix \(A\) has dimensions \(2 × 2\) and matrix \(B\) has dimensions \(2 × 2\). The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions \(2 × 2\).

We perform the operations outlined previously.

Example \(\PageIndex{5B}\): Multiplying Two Matrices

• Find \(AB\).
• Find \(BA\).

\[A=\begin{bmatrix}−1&2&3\\ 4&0&5\end{bmatrix} \nonumber\]

\[B=\begin{bmatrix}5&−1\\-4&0\\2&3\end{bmatrix} \nonumber\]

• As the dimensions of \(A\) are \(2 \times 3\) and the dimensions of \(B\) are \(3 \times 2\),these matrices can be multiplied together because the number of columns in \(A\) matches the number of rows in \(B\). The resulting product will be a \(2 \times 2\) matrix, the number of rows in \(A\) by the number of columns in \(B\).

\[ \begin{align}AB&=\begin{bmatrix}−1&2&3\\4&0&5\end{bmatrix} \begin{bmatrix}5&−1\\−4&0\\2&3\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−1(5)+2(−4)+3(2)&−1(−1)+2(0)+3(3)\\4(5)+0(−4)+5(2)&4(−1)+0(0)+5(3)\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−7&10\\30&11\end{bmatrix} \nonumber \end{align} \nonumber\]

• The dimensions of \(B\) are \(3 \times 2\) and the dimensions of \(A\) are \(2 \times 3\). The inner dimensions match so the product is defined and will be a \(3 \times 3\) matrix.

\[ \begin{align}BA&=\begin{bmatrix}5&−1\\−4&0\\2&3\end{bmatrix} \begin{bmatrix} −1&2&3\\4&0&5\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}5(−1)+−1(4)&5(2)+−1(0)&5(3)+−1(5)\\−4(−1)+0(4)&−4(2)+0(0)&−4(3)+0(5)\\2(−1)+3(4)& 2(2)+3(0)&2(3)+3(5)\end{bmatrix} \nonumber \\[4pt] &=\begin{bmatrix}−9&10&10\\4&−8&−12\\10&4&21\end{bmatrix} \nonumber \end{align} \nonumber\]

Notice that the products \(AB\) and \(BA\) are not equal.

\[AB=\begin{bmatrix}−7&10\\30&11\end{bmatrix}≠ \begin{bmatrix}−9&10&10\\4&−8&−12\\10&4&21\end{bmatrix}=BA \nonumber\]

This illustrates the fact that matrix multiplication is not commutative.

Q&A: Is it possible for AB to be defined but not BA?

Yes, consider a matrix \(A\) with dimension \(3 × 4\) and matrix \(B\) with dimension \(4 × 2\). For the product \(AB\) the inner dimensions are \(4\) and the product is defined, but for the product \(BA\) the inner dimensions are \(2\) and \(3\) so the product is undefined.

Example \(\PageIndex{6}\): Using Matrices in Real-World Problems

Let’s return to the problem presented at the opening of this section. We have Table \(\PageIndex{3}\), representing the equipment needs of two soccer teams.

We are also given the prices of the equipment, as shown in Table \(\PageIndex{4}\).

We will convert the data to matrices. Thus, the equipment need matrix is written as

\[E=\begin{bmatrix}6&10\\30&24\\14&20\end{bmatrix} \nonumber\]

The cost matrix is written as

\[C=\begin{bmatrix}300&10&30\end{bmatrix} \nonumber\]

We perform matrix multiplication to obtain costs for the equipment.

\[ \begin{align} CE&=\begin{bmatrix}300&10&30\end{bmatrix}⋅\begin{bmatrix}6&10\\30&24\\14&20\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}300(6)+10(30)+30(14)&300(10)+10(24)+30(20)\end{bmatrix} \nonumber \\[4pt] &= \begin{bmatrix}2,520&3,840\end{bmatrix} \nonumber \end{align} \nonumber\]

The total cost for equipment for the Wildcats is \($2,520\), and the total cost for equipment for the Mud Cats is \($3,840\).

How to: Given a matrix operation, evaluate using a calculator

• Save each matrix as a matrix variable \([A], [B], [C],...\)
• Enter the operation into the calculator, calling up each matrix variable as needed.
• If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.

Example \(\PageIndex{7}\): Using a Calculator to Perform Matrix Operations

Find \(AB−C\) given

\(A=\begin{bmatrix}−15&25&32\\41&−7&−28\\10&34&−2\end{bmatrix}\), \(B=\begin{bmatrix}45&21&−37\\−24&52&19\\6&−48&−31\end{bmatrix}\), and \(C=\begin{bmatrix}−100&−89&−98\\25&−56&74\\−67&42&−75\end{bmatrix}\)

On the matrix page of the calculator, we enter matrix \(A\) above as the matrix variable \([ A ]\), matrix \(B\) above as the matrix variable \([ B ]\), and matrix \(C\) above as the matrix variable \([ C ]\).

On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.

\[[A]×[B]−[C] \nonumber\]

The calculator gives us the following matrix.

\[\begin{bmatrix}−983&−462&136\\1,820&1,897&−856\\−311&2,032&413\end{bmatrix} \nonumber\]

Access these online resources for additional instruction and practice with matrices and matrix operations.

• Dimensions of a Matrix
• Matrix Addition and Subtraction
• Matrix Operations
• Matrix Multiplication

Key Concepts

• A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
• The dimensions of a matrix refer to the number of rows and the number of columns. A \(3×2\) matrix has three rows and two columns. See Example \(\PageIndex{1}\).
• We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. See Example \(\PageIndex{2}\), Example \(\PageIndex{3}\), Example \(\PageIndex{4}\), and Example \(\PageIndex{5}\).
• Scalar multiplication involves multiplying each entry in a matrix by a constant. See Example \(\PageIndex{6}\).
• Scalar multiplication is often required before addition or subtraction can occur. See Example \(\PageIndex{7}\).
• Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
• The product of two matrices, \(A\) and \(B\),is obtained by multiplying each entry in row 1 of \(A\) by each entry in column 1 of \(B\); then multiply each entry of row 1 of \(A\) by each entry in columns 2 of \(B\),and so on. See Example \(\PageIndex{8}\) and Example \(\PageIndex{9}\).
• Many real-world problems can often be solved using matrices. See Example \(\PageIndex{10}\).
• We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See Example \(\PageIndex{11}\).

• IIT JEE Study Material
• Types Of Matrices
• Types of Matrices

There are different types of matrices, and they are basically categorised on the basis of the value of their elements, their order, the number of rows and columns, etc. Now, using different conditions, the various matrix types are categorised below, along with their definition and examples.

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All Contents in Matrices

• Introduction to Matrices
• Matrix Operations
• Adjoint and Inverse of a Matrix
• Rank of a Matrix and Special Matrices
• Solving Linear Equations Using Matrix

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Matrix Types: Overview

The different types of matrices are given below:

Types of Matrices: Explanations

A matrix having only one row is called a row matrix . Thus A = [a ij ] mxn is a row matrix if m = 1. So, a row matrix can be represented as A = [a ij ] 1×n . It is called so because it has only one row, and the order of a row matrix will hence be 1 × n. For example, A = [1 2 4 5] is a row matrix of order 1 x 4. Another example of the row matrix is P = [ -4 -21 -17 ] which is of the order 1×3.

Column Matrix

A matrix having only one column is called a column matrix . Thus, A = [a ij ] mxn is a column matrix if n = 1. So, the value of a column matrix will be 1. Hence, the order is m × 1.

An example of a column matrix is:

Just like the row matrices had only one row, column matrices have only one column. Thus, the value of a column matrix will be 1. Hence, the order is m × 1. The general form of a column matrix is given by A = [a ij ] m×1 . Other examples of a column matrix include:

In the above example, P and Q are 3 ×1 and 5 × 1 order matrices, respectively.

Zero or Null Matrix

If all the elements are zero in a matrix, then it is called a zero matrix and generally denoted by 0. Thus, A = [a ij ] mxn is a zero-matrix if a ij = 0 for all i and j; E.g. \(\begin{array}{l}\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right]\end{array} \) . is a zero matrix of order 2 x 3.

is a 3 x 2 null matrix and \(\begin{array}{l}B = \left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right]\end{array} \) is 3 x 3 null matrix.

Singleton Matrix

If there is only one element in a matrix, it is called a singleton matrix . Thus, A = [a ij ] mxn is a singleton matrix if m = n = 1. E.g. [2], [3], [a], [] are singleton matrices.

Horizontal Matrix

Vertical matrix, square matrix.

If the number of rows and the number of columns in a matrix are equal, then it is called a square matrix .

is a square matrix of order 3 × 3.

Another example of a square matrix is:

The order of P and Q is 2 ×2 and 3 × 3, respectively.

Diagonal Matrix

If all the elements, except the principal diagonal, in a square matrix, are zero, it is called a diagonal matrix . Thus, a square matrix A = [a ij ] is a diagonal matrix if a ij = 0,when i ≠ j.

(i) A diagonal matrix is always a square matrix

(ii) The diagonal elements are characterized by this general form: a ij where i = j. This means that a matrix can have only one diagonal.

A few more examples of a diagonal matrix are:

In the above examples, P, Q, and R are diagonal matrices with orders 1 × 1, 2 × 2 and 3 × 3, respectively. When all the diagonal elements of a diagonal matrix are the same, it goes by a different name, the scalar matrix, which is explained below.

Scalar Matrix

If all the elements in the diagonal of a diagonal matrix are equal, it is called a scalar matrix . Thus, a square matrix \(\begin{array}{l}A={{[{{a}_{ij}}]}_{m\times m}}\ \text{is a scalar matrix if}\ a_{ij}=\left\{ \begin{matrix} 0, & i\ne j \\ k, & i=j \\ \end{matrix}\right\}\end{array} \) where k is a constant.

More examples of scalar matrices are:

Now, what if all the diagonal elements are equal to 1? That will still be a scalar matrix and obviously a diagonal matrix. It has got a special name which is known as the identity matrix .

Unit Matrix or Identity Matrix

If all the elements of a principal diagonal in a diagonal matrix are 1, it is called a unit matrix . A unit matrix of order n is denoted by I n . Thus, a square matrix A = [a ij ] m×n is an identity matrix if

Conclusions:

• All identity matrices are scalar matrices
• All scalar matrices are diagonal matrices
• All diagonal matrices are square matrices

It should be noted that the converse of the above statements is not true for any of the cases.

Equal Matrices

Equal matrices are those matrices which are equal in terms of their elements. The conditions for matrix equality are discussed below.

• Equality of Matrices Conditions

Two matrices A and B are said to be equal if they are of the same order and their corresponding elements are equal, i.e. two matrices A = [a ij ] m×n and B = [b ij ] r×s are equal if:

(a) m = r, i.e., the number of rows in A = the number of rows in B.

(b) n = s, i.e. the number of columns in A = the number of columns in B

(c) a ij = b ij , for i = 1, 2, ….., m and j = 1, 2, ….., n, i.e. the corresponding elements are equal;

For example, Matrices \(\begin{array}{l}\begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix} and \begin{bmatrix} 0 &0 & 0\\ 0& 0 &0 \end{bmatrix}\end{array} \) . are not equal because their orders are not the same.

But, If \(\begin{array}{l}A = \begin{bmatrix} 1 &6 &3\\ 5& 2&1 \end{bmatrix} and \begin{bmatrix} a_1&a_2 & a_3\\ b_1& b_2 &b_3 \end{bmatrix}\end{array} \) are equal matrices then,

a 1 = 1, a 2 = 6, a 3 = 3, b 1 = 5, b 2 = 2, b 3 = 1.

Triangular Matrix

A square matrix is said to be a triangular matrix if the elements above or below the principal diagonal are zero, and there are of two types:

• Upper Triangular Matrix

A square matrix [a ij ] is called an upper triangular matrix , if a ij = 0, when i > j.

• Lower Triangular Matrix

A square matrix is called a lower triangular matrix , if a ij = 0 when i < j.

Singular Matrix and Non-Singular Matrix

Matrix A is said to be a singular matrix if it’s determinant |A| = 0; otherwise, a non-singular matrix , i.e. if for det |A| = 0, it is singular matrix and for det |A| ≠ 0, it is non-singular.

Symmetric and Skew Symmetric Matrices

• Symmetric matrix: A square matrix A = [a ij ] is called a symmetric matrix if a ij = a ji , for all i,j values;

is symmetric, because a 12 = 2 = a 21 , a 31 = 3 = a 13 etc.

Note: A is symmetric if A’ = A (where ‘A’ is the transpose of the matrix)

• Skew-Symmetric Matrix: A square matrix A = [a ij ] is a skew-symmetric matrix if a ij = a ji , for all values of i,j.

are skew-symmetric matrices.

Note: A square matrix A is a skew-symmetric matrix A’ = -A.

Some Important Conclusions on Symmetric and Skew-Symmetric Matrices

• If A is any square matrix, then A + A’ is a symmetric matrix and A – A’ is a skew-symmetric matrix.
• Every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix. \(\begin{array}{l}A=\frac{1}{2}(A+A’)+\frac{1}{2}(A-A’)=\frac{1}{2}(B+C),\end{array} \) where B is symmetric, and C is a skew-symmetric matrix.
• If a and B are symmetric matrices, then AB is symmetric AB = BA, i.e., A & B commute.
• The matrix B’AB is symmetric or skew-symmetric in correspondence if A is symmetric or skew-symmetric.
• All positive integral powers of a symmetric matrix are symmetric.
• Positive odd integral powers of a skew-symmetric matrix are skew-symmetric, and positive even integral powers of a skew-symmetric matrix are symmetric.

Hermitian and Skew-Hermitian Matrices

A square matrix A = [a ij ] is said to be a Hermitian matrix if \(\begin{array}{l}{{a}_{ij}}={{\overline{a}}_{ji}}\,\forall \,i,j;\,i.e.\,A={{A}^{\theta }}\end{array} \) \(\begin{array}{l}E.g.\left[ \begin{matrix} a & b+ic \\ b-ic & d \\ \end{matrix} \right].\left[ \begin{matrix} 3 & 3-4i & 5+2i \\ 3+4i & 5 & -2+i \\ 5-2i & -2-i & 2 \\ \end{matrix} \right]\end{array} \) are Hermitian matrices

Important Notes:

• If A is a Hermitian matrix, then \(\begin{array}{l}{{a}_{ii}}={{\overline{a}}_{ii}}\Rightarrow {{a}_{ii}}\,is\,real\,\forall \,i,\end{array} \) thus every diagonal element of a Hermitian Matrix must be real.
• If a Hermitian matrix over the set of real numbers is actually a real symmetric matrix; and A a square matrix, A = [a ij ] is said to be a skew-Hermitian if \(\begin{array}{l}{{a}_{ij}}=-{{\overline{a}}_{ji}},\,\forall \,i,j;\end{array} \)

i.e., A θ = – A;

are skew-Hermitian matrices.

• If A is a skew-Hermitian matrix then \(\begin{array}{l}{{a}_{ii}}=-{{\overline{a}}_{ii}}\Rightarrow a_{ii}+\overline{a_{ii}}=0\end{array} \)

i.e., a ii must be purely imaginary or zero.

• A skew-Hermitian matrix over the set of real numbers is actually a real skew-symmetric matrix.

Special Matrices

(a) Idempotent Matrix:

For an idempotent matrix A, det A = 0 or x.

(b) Nilpotent Matrix:

A nilpotent matrix is said to be nilpotent of index p, \(\begin{array}{l}\left( p\in N \right),\;\; if \;\;{{A}^{p}}=O,\,\,{{A}^{p-1}}\ne O,\end{array} \) , i.e. if p is the least positive integer for which A p = O, then A is said to be nilpotent of index p.

(c) Periodic Matrix:

A square matrix which satisfies the relation A k + 1 = A, for some positive integer K, then A is periodic with period K, i.e. if K is the least positive integer for which A k + 1 = A, and A is said to be periodic with period K. If K =1, then A is called idempotent.

has period 1.

(i) Period of a square null matrix is not defined.

(ii) Period of an idempotent matrix is 1.

(d) Involutory Matrix:

If A 2 = I, the matrix is said to be an involutory matrix. An involutory matrix with its own inverse.

Matrices and Determinants – Important Topics

Matrices and Determinants – Important Questions

Frequently Asked Questions

What do you mean by a symmetric matrix.

A symmetric matrix is defined as a square matrix which is equal to its transpose matrix. We denote the transpose of a matrix A as A T . A symmetric matrix A will satisfy the condition A T = A.

What do you mean by identity matrix?

An identity matrix is a square matrix whose diagonal elements are 1, and other elements are zero. The identity matrix is also called the unit matrix.

What do you mean by Hermitian matrix?

If a square matrix is equal to its conjugate transpose matrix, then the matrix is called a Hermitian matrix.

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4.3: Types of Assignments

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• Page ID 133160

• Ana Stevenson
• James Cook University via James Cook University

Introduction

As discussed in the previous chapter, assignments are a common method of assessment at university. You may encounter many assignments over your years of study, yet some will look quite different from others. By recognising different types of assignments and understanding the purpose of the task, you can direct your writing skills effectively to meet task requirements. This chapter draws on the skills from the previous chapter, and extends the discussion, showing you where to aim with different types of assignments.

The chapter begins by exploring the popular essay assignment, with its two common categories, analytical and argumentative essays. It then examines assignments requiring case study responses , as often encountered in fields such as health or business. This is followed by a discussion of assignments seeking a report (such as a scientific report) and reflective writing assignments, which are common in nursing, education, and human services. The chapter concludes with an examination of annotated bibliographies and literature reviews. The chapter also has a selection of templates and examples throughout to enhance your understanding and improve the efficacy of your assignment writing skills.

Different Types of Written Assignments

At university, an essay is a common form of assessment. In the previous chapter Writing Assignments, we discussed what was meant by showing academic writing in your assignments. It is important that you consider these aspects of structure, tone, and language when writing an essay.

Components of an essay

Essays should use formal but reader-friendly language and have a clear and logical structure. They must include research from credible academic sources such as peer reviewed journal articles and textbooks. This research should be referenced throughout your essay to support your ideas (see the chapter Working with Information).

If you have never written an essay before, you may feel unsure about how to start. Breaking your essay into sections and allocating words accordingly will make this process more manageable and will make planning the overall essay structure much easier.

• An essay requires an introduction, body paragraphs, and a conclusion.
• Generally, an introduction and conclusion are each approximately 10% of the total word count.
• The remaining words can then be divided into sections and a paragraph allowed for each area of content you need to cover.
• Use your task and criteria sheet to decide what content needs to be in your plan

An effective essay introduction needs to inform your reader by doing four basic things:

An effective essay body paragraph needs to:

An effective essay conclusion needs to:

Common types of essays

You may be required to write different types of essays, depending on your study area and topic. Two of the most commonly used essays are analytical and argumentative . The task analysis process discussed in the previous chapter Writing Assignments will help you determine the type of essay required. For example, if your assignment question uses task words such as analyse, examine, discuss, determine, or explore, then you would be writing an analytical essay . If your assignment question has task words such as argue, evaluate, justify, or assess, then you would be writing an argumentative essay . Regardless of the type of essay, your ability to analyse and think critically is important and common across genres.

Analytical essays

These essays usually provide some background description of the relevant theory, situation, problem, case, image, etcetera that is your topic. Being analytical requires you to look carefully at various components or sections of your topic in a methodical and logical way to create understanding.

The purpose of the analytical essay is to demonstrate your ability to examine the topic thoroughly. This requires you to go deeper than description by considering different sides of the situation, comparing and contrasting a variety of theories and the positives and negatives of the topic. Although your position on the topic may be clear in an analytical essay, it is not necessarily a requirement that you explicitly identify this with a thesis statement. In an argumentative essay, however, it is necessary that you explicitly identify your position on the topic with a thesis statement. If you are unsure whether you are required to take a position, and provide a thesis statement, it is best to check with your tutor.

Argumentative essays

These essays require you to take a position on the assignment topic. This is expressed through your thesis statement in your introduction. You must then present and develop your arguments throughout the body of your assignment using logically structured paragraphs. Each of these paragraphs needs a topic sentence that relates to the thesis statement. In an argumentative essay, you must reach a conclusion based on the evidence you have presented.

Case study responses

Case studies are a common form of assignment in many study areas and students can underperform in this genre for a number of key reasons.

Students typically lose marks for not:

• Relating their answer sufficiently to the case details.
• Applying critical thinking.
• Writing with clear structure.
• Using appropriate or sufficient sources.
• Using accurate referencing.

When structuring your response to a case study, remember to refer to the case. Structure your paragraphs similarly to an essay paragraph structure, but include examples and data from the case as additional evidence to support your points (see Figure 68). The colours in the sample paragraph below show the function of each component.

The Nursing and Midwifery Board of Australia (NMBA) Code of Conduct and Nursing Standards (2018) play a crucial role in determining the scope of practice for nurses and midwives. A key component discussed in the code is the provision of person-centred care and the formation of therapeutic relationships between nurses and patients (NMBA, 2018). This ensures patient safety and promotes health and wellbeing (NMBA, 2018). The standards also discuss the importance of partnership and shared decision-making in the delivery of care (NMBA, 2018, 4). Boyd and Dare (2014) argue that good communication skills are vital for building therapeutic relationships and trust between patients and care givers. This will help ensure the patient is treated with dignity and respect and improve their overall hospital experience. In the case, the therapeutic relationship with the client has been compromised in several ways. Firstly, the nurse did not conform adequately to the guidelines for seeking informed consent before performing the examination as outlined in principle 2.3 (NMBA, 2018). Although she explained the procedure, she failed to give the patient appropriate choices regarding her health care.

Topic sentence | Explanations using paraphrased evidence including in-text references | Critical thinking (asks the so what? question to demonstrate your student voice). | Relating the theory back to the specifics of the case. The case becomes a source of examples as extra evidence to support the points you are making.

Reports are a common form of assessment at university and are also used widely in many professions. It is a common form of writing in business, government, scientific, and technical occupations.

Reports can take many different structures. A report is normally written to present information in a structured manner, which may include explaining laboratory experiments, technical information, or a business case. Reports may be written for different audiences, including clients, your manager, technical staff, or senior leadership within an organisation. The structure of reports can vary, and it is important to consider what format is required. The choice of structure will depend upon professional requirements and the ultimate aims of the report. Consider some of the options in the table below (see Table 18.2).

Reflective writing

Reflective writing is a popular method of assessment at university. It is used to help you explore feelings, experiences, opinions, events, or new information to gain a clearer and deeper understanding of your learning.

A reflective writing task requires more than a description or summary. It requires you to analyse a situation, problem or experience, consider what you may have learnt, and evaluate how this may impact your thinking and actions in the future. This requires critical thinking, analysis, and usually the application of good quality research, to demonstrate your understanding or learning from a situation.

Essentially, reflective practice is the process of looking back on past experiences and engaging with them in a thoughtful way and drawing conclusions to inform future experiences. The reflection skills you develop at university will be vital in the workplace to assist you to use feedback for growth and continuous improvement. There are numerous models of reflective writing and you should refer to your subject guidelines for your expected format. If there is no specific framework, a simple model to help frame your thinking is What? So what? Now what? (Rolfe et al., 2001).

The Gibbs’ Reflective Cycle

The Gibbs’ Cycle of reflection encourages you to consider your feelings as part of the reflective process. There are six specific steps to work through. Following this model carefully and being clear of the requirements of each stage, will help you focus your thinking and reflect more deeply. This model is popular in Health.

The 4 R’s of reflective thinking

This model (Ryan and Ryan, 2013) was designed specifically for university students engaged in experiential learning. Experiential learning includes any ‘real-world’ activities, including practice led activities, placements, and internships. Experiential learning, and the use of reflective practice to heighten this learning, is common in Creative Arts, Health, and Education.

Annotated bibliography

What is it.

An annotated bibliography is an alphabetical list of appropriate sources (e.g. books, journal articles, or websites) on a topic, accompanied by a brief summary, evaluation, and sometimes an explanation or reflection on their usefulness or relevance to your topic. Its purpose is to teach you to research carefully, evaluate sources and systematically organise your notes. An annotated bibliography may be one part of a larger assessment item or a stand-alone assessment item. Check your task guidelines for the number of sources you are required to annotate and the word limit for each entry.

How do I know what to include?

When choosing sources for your annotated bibliography, it is important to determine:

• The topic you are investigating and if there is a specific question to answer.
• The type of sources on which you need to focus.
• Whether these sources are reputable and of high quality.

What do I say?

Important considerations include:

• Is the work current?
• Is the work relevant to your topic?
• Is the author credible/reliable?
• Is there any author bias?
• The strength and limitations (this may include an evaluation of research methodology).

Literature reviews

Generally, a literature review requires that you review the scholarly literature and establish the main ideas that have been written about your chosen topic. A literature review does not summarise and evaluate each resource you find (this is what you would do in an annotated bibliography). You are expected to analyse and synthesise or organise common ideas from multiple texts into key themes which are relevant to your topic (see Figure 18.10). You may also be expected to identify gaps in the research.

It is easy to get confused by the terminology used for literature reviews. Some tasks may be described as a systematic literature review when actually the requirement is simpler; to review the literature on the topic but do it in a systematic way. There is a distinct difference (see Table 15.4). As a commencing undergraduate student, it is unlikely you would be expected to complete a systematic literature review as this is a complex and more advanced research task. It is important to check with your lecturer or tutor if you are unsure of the requirements.

When conducting a literature review, use a table or a spreadsheet, if you know how, to organise the information you find. Record the full reference details of the sources as this will save you time later when compiling your reference list (see Table 18.5).

Overall, this chapter has provided an introduction to the types of assignments you can expect to complete at university, as well as outlined some tips and strategies with examples and templates for completing them. First, the chapter investigated essay assignments, including analytical and argumentative essays. It then examined case study assignments, followed by a discussion of the report format. Reflective writing , popular in nursing, education, and human services, was also considered. Finally, the chapter briefly addressed annotated bibliographies and literature reviews. The chapter also has a selection of templates and examples throughout to enhance your understanding and improve the efficacy of your assignment writing skills.

• Not all assignments at university are the same. Understanding the requirements of different types of assignments will assist in meeting the criteria more effectively.
• There are many different types of assignments. Most will require an introduction, body paragraphs, and a conclusion.
• An essay should have a clear and logical structure and use formal but reader-friendly language.
• Breaking your assignment into manageable chunks makes it easier to approach.
• Effective body paragraphs contain a topic sentence.
• A case study structure is similar to an essay, but you must remember to provide examples from the case or scenario to demonstrate your points.
• The type of report you may be required to write will depend on its purpose and audience. A report requires structured writing and uses headings.
• Reflective writing is popular in many disciplines and is used to explore feelings, experiences, opinions, or events to discover what learning or understanding has occurred. Reflective writing requires more than description. You need to be analytical, consider what has been learnt, and evaluate the impact of this on future actions.
• Annotated bibliographies teach you to research and evaluate sources and systematically organise your notes. They may be part of a larger assignment.
• Literature reviews require you to look across the literature and analyse and synthesise the information you find into themes.

Gibbs, G. (1988). Learning by doing: A guide to teaching and learning methods. Further Education Unit, Oxford Brookes University.

Rolfe, G., Freshwater, D., Jasper, M. (2001). Critical reflection in nursing and the helping professions: A user’s guide . Palgrave Macmillan.

Ryan, M. & Ryan, M. (2013). Theorising a model for teaching and assessing reflective learning in higher education. Higher Education Research & Development , 32(2), 244-257. https://doi.org/10.1080/07294360.2012.661704

Writing Guide: Types of Assignments & Best Practices

• Home & Appointments
• Types of Assignments & Best Practices
• Tables & Figures
• Thesis & Project Guide

The most common types of writing assignments you will encounter at MLTS

• How to approach a writing assignment
• Expository writing & research papers
• Compare & Contrast paper
• Book & Literature Reviews
• Reflective writing
• Online discussion posts
• Thesis/Project

As a graduate student, you will be assigned a variety of types of writing projects.  A good rule of thumb in approaching any writing project is to ask yourself: for whom am I writing and why?  Or, who is my audience and what do they expect from my writing?  Your assignments will almost invariably require you to make one or more arguments.   A good argument is well-written, logical, and supported by evidence.

Expository writing involves understanding, explaining, analyzing, and/or evaluating a topic.  It includes your standard graduate school essay, book review, or research paper where your instructor requires you to analyze and/or study a topic.  In general, your audience for such assignments will be your course instructor. You can think of such writing assignments as your instructor asking you to make an argument.  Your instructor wants to gauge your creative thinking skills and how well you understand the course material by seeing how well you can make an argument related to that material.  Remember: a good argument is well-written, logical, and supported by evidence.

An expository paper is therefore not about you (at least not directly); it is about the facts you have learned and researched and the argument you have built from those facts. Therefore, unless you are quoting someone, you should avoid using first person pronouns (the words I, me, my, we, us, our ) in your writing.  Let your facts and arguments speak for themselves instead of beginning statements with "I think" or "I believe."

A compare & contrast assignment is a type of expository & research paper assignment.  It is important to organize your writing around the themes you are comparing & contrasting.  If, for example, you are assigned to compare & contrast, say, Augustine's Confessions and The Autobiography of Malcolm X , a common mistake students make is to write the first part of their essay strictly about Augustine's Confessions , and the second part of the essay strictly about The Autobiography of Malcolm X .  In a good compare & contrast essay, you instead explore an issue in every paragraph or two, and show how, in this case, both Augustine & Malcolm X share common ground or differ on that issue.  Then, move onto another issue and show how both Augustne and Malcolm X covered it.

Unless your instructor directs you otherwise, you should not use first person pronouns ( I, me, my ) in such a paper.

A book review assignment is meant to be an analysis of a book, not a chapter-by-chapter summary of a book.  Instead of organizing your paper sequentially (the first paragraph is about chapter 1, the second paragraph is about chapter 2, etc.), organize your paragraphs around the themes of the book that are thread throughout the book.  Topics to consider in a book review include (but are not limited to):

• What are the author's arguments, and how successful is she in making those arguments?
• What sort of sources does the author utilize?
• What methodology/methodologies does the author utilize?
• What are the strengths and weaknesses of the book?

A literature review is similar to a book review assignment in that it is meant to be an analysis of a theme or themes across several books/articles.   What have various authors written about your topic?   That said, as you will typically have less space to talk about each work (perhaps a paragraph or less for each work as opposed to multiple pages), you might end up moving from one author's findings to another.  For a literature review in a thesis, think of a literature review as a mini-essay within your broader thesis with its own mini-introduction, thesis statement, and conclusion.

Unless your instructor directs you otherwise, book reviews and literature reviews should be written like expository & research papers.  In particular, you should not use first person pronouns ( I, me, my ).  So, instead of writing: "I think this book is a good analysis of ___," write: "This book is a good analysis of ___."

Reflective essays are especially common in theology courses.  Reflective writing requires that you explicitly write about yourself and your own views.  To put it another way, you typically have two audiences to write for in such an assignment: your instructor and yourself.   As such, and unlike a standard expository paper, such essays require you to write about yourself using first person pronouns ( I, me, my) and use statements like “I think” and “I believe.”  Otherwise, a reflective essay shares a lot with expository writing.  You are still making arguments, and you still need evidence from cited sources!  Unless your instructor tells you otherwise, you should still include a good title, introduction paragraph, thesis statement, conclusion, and bibliography.

For online courses, you will likely have to take part in classroom or group discussions online, in which you will be encouraged or even required to respond to your classmates.  Such writing assignments often include a reflective element. Discussion posts are almost always shorter than essays and as such may not need long introductions or conclusions.  That said, a discussion post is not like a Facebook or social media post!  Good discussion posts are long and well-written enough to convey one or more thoughtful, insightful observations; you cannot just "like" someone else's post or only write "Good job!"  If you decide to challenge or critique a classmate’s post—and you are certainly encouraged to do so!—you should do so in a respectful and constructive manner.   As your main audience for online discussions are your own classmates and, to a lesser extent, your instructor, it is often okay to use relatively more informal language and to refer to yourself using first person pronouns ( I, me, my ).  Finally, as with reflective essays, discussion posts still benefit from evidence.  Even if a discussion post is relatively less formal than an essay, if you quote, paraphrase, or draw ideas from outside sources, you still must cite them!  If the online medium does not allow for footnotes, use parenthetical references for citations (see chapter 19 of Turabian).

Those of you taking preaching courses or earning a DMin degree will have to write and submit your sermons. On one hand, your main audience for such a writing assignment is the congregation to whom you may preach.  The language, tone, message, level of detail, etc. of a good sermon will depend on the precise context of your congregation and the message you want to impart.  Therefore, unlike an expository essay or a reflective essay, you have a lot more freedom in how you chose to organize your sermon, as well as how formal or not you want the language to be.

On the other hand, in submitting such assignments, you also have a secondary audience: your instructor.  As such, you may still need to include citations, even if you would not read them out loud in your sermon.  In submitting a sermon as an assignment, you may also need to include some sort of write up or commentary, which your instructor may require to be expository and/or reflective in nature.

Those of you earning an MAR or DMin will finish your coursework by proposing, researching, writing, and defending a thesis or project.   A thesis/project should be an original contribution to your field of study.  To put it another way, the audience for your thesis/project is not just your advisor, but the broader academic and/or ministerial community.   A good thesis/project can go on to become the first draft of a published academic journal article or a chapter or two of a book.  Your thesis/project should be largely expository, but it may also include reflective sections.

It is never too early to start thinking about what you want to do for your thesis/project! You can try to make your thesis/project writing process easier by writing your course papers on topics within or adjacent to what you think you want to do for your thesis/project; that said, if you do so, you will need to cite these earlier works in your final thesis.   See our citation guide for help with that.

For more information on writing a thesis or project, from choosing a topic to submitting it, check out our Thesis & Project Guide .

Tips for Composing Good Academic Prose

• Proofread, proofread, proofread!
• Find evidence to support your thesis statement from good quality sources
• Use quotations as evidence, not filler
• Be careful not to turn long sentences into run-on sentences
• Relatively longer paragraphs are generally better than short ones
• Make sure your paper flows well from one idea to the next
• When possible, avoid using the passive voice
• Be precise and crystal-clear in your statements and arguments
• Use the present tense when paraphrasing an author or setting up a quotation
• Use repetition of words carefully

First and most importantly: Proofread your paper over before you submit it to make sure that it reads well and is without errors!  Read your paper over as you are writing it.  Check over your work with spell check.  Before you submit it, read it over one last time to catch anything you missed.  If possible, consider reading the work out loud: you will be more likely to spot problems in your writing than if you read it in your head.  If you are able to do so, ask a friend or schedule an appointment with the Writing Center for a review.  Another pair of eyes can often spot a mistake or problem that the writer has overlooked.

Find evidence to support your thesis from good quality sources.   Your research and writing should be based on the study of reputable primary and secondary sources.  Typically, this means books published by academic presses and academic journal articles.  Wikipedia, YouTube, random websites, and dictionary entries are generally not considered to be good sources for academic writing, although there are instances when it is acceptable to use and cite them, like if you were researching how topics in Black theology are represented or misrepresented on Wikipedia.  If you need help in finding good resources for your paper, consult a librarian.

Quotations are meant to be evidence to support your argument; they are not filler to meet a length requirement.   While you must quote and paraphrase sources, you should not quote or paraphrase more than you need.  When possible, consider paraphrasing over quoting.  Keep in mind that your writing assignments are supposed to showcase your thinking and writing, not the thinking and writing of whoever you are citing.

Be careful not to turn long sentences into run-on sentences.   Long sentences are not always bad: when well-written, a long sentence can read better and help convey complex ideas better than a series of short sentences.  A run-on sentence, on the other hand, occurs when multiple sentences are inappropriately lumped into a single sentence.  Therefore, when reading your paper over, keep an eye out for any sentence that you can break into multiple sentences.

Relatively longer paragraphs are generally better than short ones.   If your paragraph is three sentences or less, consider if you can write more about that paragraph's topic or incorporate it into another paragraph.  If a paragraph represents one idea, then a longer paragraph typically shows that you have better considered and flushed out that idea.  That said, if your paragraph is longer than a page, you could probably shorten it or break it into two paragraphs.

Make sure your paper flows well from one idea to the next.   Does your third paragraph make sense following your second paragraph?  Do you drop ideas and only pick them up much later?  Cut and paste sentences and paragraphs around as necessary.

When possible, avoid using the passive voice.   This can be tricky!  The passive voice is when you use the verb “to be” next to and in conjunction with another verb to make the object of the sentence into the subject.  For example, compare the active sentence: “Kate Turabian wrote the book” to its passive equivalent: “The book was written by Kate Turabian.”  Grammatically speaking, in the latter, passive sentence, "The book" is the subject, even though in a real world active sense, it is the object.

Writers consider passive sentences not as good because, like in the above example, they can be wordier than necessary and take the focus off the real subject. There are exceptions in which it is good to use the passive voice. For example, if you were writing an article about Kate Turabian, it would be better to write: “Kate Turabian was born in 1893” instead of “Kate Turabian’s mother gave birth to her in 1893.”  The former sentence keeps Kate Turabian, the focus of the paper, as the subject, while the latter sounds a little weird (maybe English speakers are too squeamish, but we typically do not recount someone's birth in that way).

At its worst, the passive voice can obscure the subject and make facts unclear. Consider the sentence: "Jackie Robinson's signing with the Brooklyn Dodgers in 1946 was considered a crucial moment in the Civil Rights movement." With the passive voice, the reader does not know who exactly considered that so?  Did all Americans in 1946 think this?  Did some specific people come to recognize it later?  Compare that sentence to: "Martin Luther King, Jr. considered Jackie Robinson's signing with the Brooklyn Dodgers in 1946 a crucial moment in the Civil Rights movement."

Be precise and crystal-clear in your statements and arguments.   Similar to how the passive voice can make facts unclear, overly general language can make for weak arguments.  Consider the argument: "Many people now support same-sex marriage."  Many people?  Which people?  "Many" and "people" are very general terms and do not tell us much in this statement; the more specific you can be, the better your argument:

• Despite official church statements, many American Catholics now support same-sex marriage.
• [Specific number]% of Chicagoans now support same-sex marriage.
• Many South African theologians, including [so-and-so] and [so-and so], now support same-sex marriage.

In general, use the present tense when paraphrasing an author or setting up a quotation.  While you should use the past tense when writing about events in the past, you should in general use the present tense when discussing a scholar's writing.  Scholarship is a ongoing discussion.  When you read and discuss an author's work, that author is making an argument right now in the present, even if she is dead.  So, do not write:

   Carl Jung wrote: "The psyche...    Carl Jung said, "The psyche...    Carl Jung argued that...

but instead:

   Carl Jung writes: "The psyche...    Carl Jung says, "The psyche...    Carl Jung argues that...

Use repetition of words carefully.   When done well, repeating words can sound good and emphasize ideas.  When done poorly, repetition sounds monotonous. Avoid, for example, starting too many sentences or paragraphs with the same word, or overutilizing the same verb.  If you need help in bringing variety to your word choices, purchase a thesaurus or check out thesaurus.com .

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Assignment Problem: Meaning, Methods and Variations | Operations Research

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table: