# Tagged: matrix power

## Problem 688

Let $A$ be a $3\times 3$ matrix and let
$\mathbf{v}=\begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}.$ Suppose that $A\mathbf{v}=-\mathbf{v}$ and $A\mathbf{w}=2\mathbf{w}$.
Then find the vector
$A^5\begin{bmatrix} -1 \\ 8 \\ -9 \end{bmatrix}.$