Question:
If matrix $A=\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]$ and $A^{2}=p A$, then write the value of $p$.
Solution:
Given: $A=\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]$
$A^{2}=\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]$
$=\left[\begin{array}{cc}4+4 & -4-4 \\ -4-4 & 4+4\end{array}\right]$
$=\left[\begin{array}{cc}8 & -8 \\ -8 & 8\end{array}\right]$
$=4\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]$
$=4 A$
Hence, $p=4$.