# Identity Matrix

How to use the definition of the identity matrix to prove the given statement: definition, 1 example, and its solution.

## Definition

The identity matrix is a matrix

that satisfies

AI = IA = A.

The identity matrix is a square matrix.

(number of rows = number of columns)

The diagonal elements are 1.

And the other elements are 0.

## Example

(A + I)^{2} = (A + I)(A + I)

Solve (A + I)(A + I)

by using the FOIL method.

Multiply the first two matrices:

AA = A^{2}.

Multiply the outer matrices:

+AI.

Multiply the inner matrices:

+IA.

Multiply the last two matrices:

+II = +I^{2}.

+AI = +A

+IA = +A

+I^{2} = +II = +I

+A + A = +2A

Let's see what you've solved.

You changed the left side, (A + I)^{2},

to the right side, A^{2} + 2A + I.

So

(A + I)^{2} = A^{2} + 2A + I

is true.

So the given statement,

(A + I)^{2} = A^{2} + 2A + I,

is true.