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

Kyran Hudson
2021-03-12
Answered

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

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smallq9

Answered 2021-03-13
Author has **106** answers

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,
$A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

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$2\times 2$ . 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$2\times 2$ 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.

${C}^{2}=C$

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right],\text{and}\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$

Thus, three idempotent matrices are

Given data:

The order of the given matrix is:

The expression of the identity matrix of

The expression for the square of the given matrix is,

Substitute the given matrix in the above expression.

Step 2

Assume the given matrix of order

The expression for the square of the given matrix is,

Substitute the value in the expression.

Step 3

Assume the given matrix of order

The expression for the square of the given matrix is,

Substitute the value in the expression.

Thus, three idempotent matrices are

Jeffrey Jordon

Answered 2022-01-24
Author has **2262** answers

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