# Find three 2 times 2 idempotent matrices. (Recall that a square matrix A is idempotent when A^2 = A)

Matrices
Find three $$2 \times 2$$ idempotent matrices. (Recall that a square matrix A is idempotent when $$A^2 = A$$)

2021-03-13
Step 1
Given data:
The order of the given matrix is: $$m \times n = 2 \times 2$$. The expression for the idempotent matrix is,
$$x^2=x$$
The expression of the identity matrix of $$2 \times 2$$ is, $$A=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}$$
The expression for the square of the given matrix is,
$$A^2=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}$$
$$=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}$$
Substitute the given matrix in the above expression.
$$A^2=A$$
Step 2
Assume the given matrix of order $$2 \times 2$$. is, $$B=\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}$$
The expression for the square of the given matrix is, $$B^2=\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}$$
$$=\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}$$
Substitute the value in the expression.
$$B^2=B$$
Step 3
Assume the given matrix of order $$2 \times 2$$ is, $$C=\begin{bmatrix}1 & 1 \\0 & 0 \end{bmatrix}$$
The expression for the square of the given matrix is, $$C^2=\begin{bmatrix}1 & 1 \\0 & 0 \end{bmatrix} \begin{bmatrix}1 & 1 \\0 & 0 \end{bmatrix}$$
$$=\begin{bmatrix}1 & 1 \\0 & 0 \end{bmatrix}$$
Substitute the value in the expression.
$$C^2=C$$
$$\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix} , \begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix} , \text{ and } \begin{bmatrix}1 &1 \\0 & 0 \end{bmatrix}$$
Thus, three idempotent matrices are