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 $A^5\mathbf{u}$.
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