To determine which matrices satisfy the condition [tex]\(A^2 = A\)[/tex], known as being idempotent, we need to verify that for each given matrix [tex]\(A\)[/tex], squaring the matrix results in the same matrix. Let's analyze each matrix step by step.
1. Matrix [tex]\(A_1\)[/tex]:
[tex]\[
A_1 = \begin{pmatrix}
5 & 5 \\
-4 & -4
\end{pmatrix}
\][/tex]
This matrix is idempotent because [tex]\(A_1^2 = A_1\)[/tex].
2. Matrix [tex]\(A_2\)[/tex]:
[tex]\[
A_2 = \begin{pmatrix}
6 & 5 \\
5 & 6
\end{pmatrix}
\][/tex]
This matrix is not idempotent because [tex]\(A_2^2 \neq A_2\)[/tex].
3. Matrix [tex]\(A_3\)[/tex]:
[tex]\[
A_3 = \begin{pmatrix}
0.5 & -0.5 \\
-0.5 & 0.5
\end{pmatrix}
\][/tex]
This matrix is idempotent because [tex]\(A_3^2 = A_3\)[/tex].
4. Matrix [tex]\(A_4\)[/tex]:
[tex]\[
A_4 = \begin{pmatrix}
0.5 & 0.5 \\
-0.5 & 0.5
\end{pmatrix}
\][/tex]
This matrix is not idempotent because [tex]\(A_4^2 \neq A_4\)[/tex].
5. Matrix [tex]\(A_5\)[/tex]:
[tex]\[
A_5 = \begin{pmatrix}
-6 & -6 \\
5 & 5
\end{pmatrix}
\][/tex]
This matrix is not idempotent because [tex]\(A_5^2 \neq A_5\)[/tex].
Therefore, the matrices which are idempotent (satisfying [tex]\(A^2 = A\)[/tex]) are:
[tex]\[
\begin{pmatrix}
5 & 5 \\
-4 & -4
\end{pmatrix}
\][/tex]
and
[tex]\[
\begin{pmatrix}
0.5 & -0.5 \\
-0.5 & 0.5
\end{pmatrix}.
\][/tex]
