If $\displaystyle \mathrm{A}=\left[\begin{array}{rr}2 & -2 \\ -2 & 2\end{array}\right]$ and $\displaystyle \mathrm{A}^{2}=\mathrm{kA}$, then find the value of k .

\[\begin{aligned} & \mathrm{A}=\left[\begin{array}{cc} 2 & -2 \\ -2 & 2 \end{array}\right] \Rightarrow \mathrm{A}^{2}=\left[\begin{array}{cc} 8 & -8 \\ -8 & 8 \end{array}\right] \\ & \mathrm{A}^{2}=k \mathrm{A} \Rightarrow\left[\begin{array}{cc} 8 & -8 \\ -8 & 8 \end{array}\right]=\left[\begin{array}{cc} 2 k & -2 k \\ -2 k & 2 k \end{array}\right] \\ & \Rightarrow k=4 \end{aligned} \]