Question

If `A=[[0,0],[4,0]]` find `A^16`

Answer 1

\[Given: A = \begin{bmatrix}0 & 0 \\ 4 & 0\end{bmatrix}\]
\[Here, \]
\[ A^2 = AA\]
\[ \Rightarrow A^2 = \begin{bmatrix}0 & 0 \\ 4 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 4 & 0\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}0 + 0 & 0 + 0 \\ 0 + 0 & 0 + 0\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ A^4 = A^2 A^2 \]
\[ \Rightarrow A^4 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \Rightarrow A^4 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ A^8 = A^4 A^4 \]
\[ \Rightarrow A^8 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \Rightarrow A^8 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ A^{16} = A^8 A^8 \]
\[ \Rightarrow A^{16} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \therefore A^{16} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
 Thus, `A^16` is a null matrix .