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

Find three $2×2$ idempotent matrices. (Recall that a square matrix A is idempotent when ${A}^{2}=A$)
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Step 1
Given data:
The order of the given matrix is: $m×n=2×2$. The expression for the idempotent matrix is,
${x}^{2}=x$
The expression of the identity matrix of $2×2$ is, $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
The expression for the square of the given matrix is,
${A}^{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
Substitute the given matrix in the above expression.
${A}^{2}=A$
Step 2
Assume the given matrix of order $2×2$. is, $B=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
The expression for the square of the given matrix is, ${B}^{2}=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
$=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
Substitute the value in the expression.
${B}^{2}=B$
Step 3
Assume the given matrix of order $2×2$ is, $C=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$
The expression for the square of the given matrix is, ${C}^{2}=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$
$=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$
Substitute the value in the expression.
${C}^{2}=C$

Thus, three idempotent matrices are
Jeffrey Jordon