Solution MM.C32 Chris Black

$A^{2} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0\\ 0 & 0 & 9\end{bmatrix}$, $A^{3} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 8 & 0\\ 0 & 0 & 27\end{bmatrix}$, and $A^{4} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 16 & 0\\ 0 & 0 & 81\end{bmatrix}$. The pattern emerges, and we see that $A^{n} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2^n & 0\\ 0 & 0 & 3^n\end{bmatrix}$.