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