Identity Matrix

Identity Matrix

How to use the identity matrix to solve matrix problems: definition, example, and its solution.

Definition

The identity matrix is a matrix that satisfy AI = IA = A. Its diagonal elements are 1. And the other elements are 0.

The identity matrix [I] is a matrix
that satisfies
AI = IA = A.

Its diagonal elements are 1.
And the other elements are 0.

Example

If A = [1 2 / 3 4], find AIA.

By the definition of the identity matrix,
AI = A.

So AIA = AA.

A = [1 2 / 3 4]

So AA = [1 2 / 3 4][1 2 / 3 4].

Multiply these two matrices.

Multiplying matrices

[Row 1]⋅[Column 1]: 1⋅1 + 2⋅3
[Row 1]⋅[Column 2]: 1⋅2 + 2⋅4

[Row 2]⋅[Column 1]: 3⋅1 + 4⋅3
[Row 2]⋅[Column 2]: 3⋅2 + 4⋅4

1⋅1 + 2⋅3 = 1 + 6
1⋅2 + 2⋅4 = 2 + 8

3⋅1 + 4⋅3 = 3 + 12
3⋅2 + 4⋅4 = 6 + 16

1 + 6 = 7
2 + 8 = 10

3 + 12 = 15
6 + 16 = 22

So AIA = [7 10 / 15 22].