Question

If $A=\left[\begin{array}{cc}-3& 0\\ 0& -3\end{array}\right]$, find A⁴.

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ {A}^2=AA \\ \Rightarrow A^2=\left[\begin{array}{cc}-3& 0\\ 0& -3\end{array}\right]\left[\begin{array}{cc}-3& 0\\ 0& -3\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}9+0& 0+0\\ 0+0& 0+9\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right] \\ \mathrm{Now}, \\ {A}^4=A^2{A}^2 \\ \Rightarrow A^4=\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right]\left[\begin{array}{cc}9& 0\\ 0& 9\end{array}\right] \\ \Rightarrow A^4=\left[\begin{array}{cc}81+0& 0+0\\ 0+0& 0+81\end{array}\right] \\ \Rightarrow A^4=\left[\begin{array}{cc}81& 0\\ 0& 81\end{array}\right] \end{gathered}$$