Question

If \[A = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\] , find A⁴.

Answer 1

\[Here, \] 

`A^2 `= AA

\[ \Rightarrow A^2 = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\] 

\[ \Rightarrow A^2 = \begin{bmatrix}9 + 0 & 0 + 0 \\ 0 + 0 & 0 + 9\end{bmatrix}\] 

\[ \Rightarrow A^2 = \begin{bmatrix}9 & 0 \\ 0 & 9\end{bmatrix}\] 

\[Now, \] 

\[ A^4 = A^2 A^2 \] 

\[ \Rightarrow A^4 = \begin{bmatrix}9 & 0 \\ 0 & 9\end{bmatrix}\begin{bmatrix}9 & 0 \\ 0 & 9\end{bmatrix}\] 

\[ \Rightarrow A^4 = \begin{bmatrix}81 + 0 & 0 + 0 \\ 0 + 0 & 0 + 81\end{bmatrix}\] 

\[ \Rightarrow A^4 = \begin{bmatrix}81 & 0 \\ 0 & 81\end{bmatrix}\]