To determine which matrices [tex]\(A\)[/tex] satisfy the equation [tex]\(A^2 = A\)[/tex], we need to go through each matrix and see if this condition holds true.
Given the matrices:
1. [tex]\(\left[\begin{array}{cc}5 & 5 \\ -4 & -4\end{array}\right]\)[/tex]
2. [tex]\(\left[\begin{array}{cc}6 & 5 \\ 5 & 6\end{array}\right]\)[/tex]
3. [tex]\(\left[\begin{array}{cc}0.5 & -0.5 \\ -0.5 & 0.5\end{array}\right]\)[/tex]
4. [tex]\(\left[\begin{array}{cc}0.5 & 0.5 \\ -0.5 & 0.5\end{array}\right]\)[/tex]
5. [tex]\(\left[\begin{array}{cc}-6 & -6 \\ 5 & 5\end{array}\right]\)[/tex]
We need to determine which of these matrices satisfy [tex]\(A^2 = A\)[/tex].
After verifying these conditions, the matrices that satisfy [tex]\(A^2 = A\)[/tex] are:
1. [tex]\(\left[\begin{array}{cc}5 & 5 \\ -4 & -4\end{array}\right]\)[/tex]
3. [tex]\(\left[\begin{array}{cc}0.5 & -0.5 \\ -0.5 & 0.5\end{array}\right]\)[/tex]
Therefore, the correct answers are:
[tex]\[
\left[\begin{array}{cc}5 & 5 \\ -4 & -4\end{array}\right]
\][/tex]
and
[tex]\[
\left[\begin{array}{cc}0.5 & -0.5 \\ -0.5 & 0.5\end{array}\right]
\][/tex]
