Solve the following equations

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, $p=4$.

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