Identity Matrix

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.

(A + I)² = (A + I)(A + I)

Solve (A + I)(A + I)
by using the FOIL method.

Multiply the first two matrices:
AA = A².

Multiply the outer matrices:
+AI.

Multiply the inner matrices:
+IA.

Multiply the last two matrices:
+II = +I².

+AI = +A
+IA = +A
+I² = +II = +I

+A + A = +2A

Let's see what you've solved.
You changed the left side, (A + I)²,
to the right side, A² + 2A + I.

So
(A + I)² = A² + 2A + I
is true.

So the given statement,
(A + I)² = A² + 2A + I,
is true.