Login

Identity Matrix: Definition & Properties

Instructor: Norair Sarkissian
In this lesson, we will learn about the identity matrix, which is a square matrix that has some unique properties. We will discover that a given matrix may have more than one identity matrix.

Definition

An identity matrix is a matrix whose product with another matrix A equals the same matrix A.

Any matrix typically has two different identity matrices: a left identity matrix and a right identity matrix. If I is a left identity matrix for a given matrix A, then the matrix product I.A = A. If I is a right identity matrix for A, then the matrix product A.I = A.

Identity matrix

The identity matrix is a square matrix which contains ones along the main diagonal (from the top left to the bottom right), while all its other entries are zero. Such a matrix is of the form given below:

null

For example, the 4-by-4 identity matrix is shown below:

null

Left and right identity matrices

The matrix product A.B is only possible if matrix A has the same number of columns as the number of rows in matrix B. Thus, if A has n columns, we can only perform the matrix multiplication A.B, if B has n rows. So in general, if A is an m-by-n matrix, then B must be an n-by-p matrix.

Applying the same concept to identity matrices, we can see that if A is an m-by-n matrix (consisting of m rows and n columns), it will have a left identity matrix which is an m-by-m square matrix, and a right identity matrix whose dimensions are n-by-n.

Dimensions of the identity matrix

It may be constructive to ask why the identity matrix I is always a square matrix. Let's assume that a matrix A is m-by-n, and that its identity matrix I is a p-by-q matrix. Now, if I is a left identity matrix, we know that I.A = A. For the matrix product to be possible, I must have as many columns as the number of rows of A. This means that q = m. Thus, I is a p-by-m matrix.

To unlock this lesson you must be a Study.com Member.
Create your account

Register for a free trial

Are you a student or a teacher?
I am a teacher

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back

Earning College Credit

Did you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it free for 5 days!
Create An Account
Support