Question
If \[ A = \begin{bmatrix} 0 & 0 \\ 4 & 0 \end{bmatrix} \] find \(A^{16}\).
Solution
Step 1: Compute \(A^2\)
\[ A^2 = \begin{bmatrix} 0 & 0 \\ 4 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 4 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]Step 2: Observe Pattern
\[ A^2 = O \Rightarrow A^n = O \quad \text{for all } n \ge 2 \]Step 3: Apply
\[ A^{16} = O \]Final Answer
\[
A^{16} =
\begin{bmatrix}
0 & 0 \\
0 & 0
\end{bmatrix}
\]