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High School Algebra II: Homework Help Resource26 chapters | 280 lessons | 1 flashcard set

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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.

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*.

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:

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*.

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.

In addition, we know that when we multiply a *p*-by-*m* matrix with an *m*-by-*n* matrix, the result is a *p*-by-*n* matrix. But since the product of *I* and *A* equals *A*, it means that the product is an *m*-by-*n* matrix. Therefore, *p* = *m* as well. Thus both *p* and *q* equal *m* and the left identity matrix is a square *m*-by-*m* matrix.

We can use similar reasoning to deduce that for an *m*-by-*n* matrix *A*, the right identity matrix *I* is a square *n*-by-*n* matrix.

For example, let us determine the left and right identity matrices for the 2-by-3 matrix *A* below:

The left identity matrix is a 2-by-2 matrix with ones in the main diagonal:

It can be verified that the matrix product *I.A* equals *A*.

The right identity matrix is a 3-by-3 matrix with ones in the main diagonal:

In this case, we can also verify that the matrix product *A.I* equals *A*.

The identity matrix is a square matrix whose product with another matrix *A* equals the same matrix *A*. The identity matrix is a square matrix which contains all zeroes, except for ones along the main diagonal.

If *A* is an *m*-by-*n* matrix, its left identity matrix is *m*-by-*m* and its right identity matrix is *n*-by-*n*.

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High School Algebra II: Homework Help Resource26 chapters | 280 lessons | 1 flashcard set

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