# Identity Matrix: Definition & Properties

Instructor: Norair Sarkissian

Norair holds master's degrees in electrical engineering and mathematics

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:

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

## 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.

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