Question

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

Matrices
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asked 2021-03-12
Find three \(2 \times 2\) idempotent matrices. (Recall that a square matrix A is idempotent when \(A^2 = A\))

Answers (1)

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
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