## Compute $A^5\mathbf{u}$ Using Linear Combination

## Problem 696

Let

\[A=\begin{bmatrix}

-4 & -6 & -12 \\

-2 &-1 &-4 \\

2 & 3 & 6

\end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix}

6 \\

5 \\

-3

\end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix}

-2 \\

0 \\

1

\end{bmatrix}, \quad \text{ and } \mathbf{w}=\begin{bmatrix}

-2 \\

-1 \\

1

\end{bmatrix}.\]

**(a)** Express the vector $\mathbf{u}$ as a linear combination of $\mathbf{v}$ and $\mathbf{w}$.

**(b)** Compute $A^5\mathbf{v}$.

**(c)** Compute $A^5\mathbf{w}$.

**(d)** Compute $A^5\mathbf{u}$.