## How to Obtain Information of a Vector if Information of Other Vectors are Given

## 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}.\]