## Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$

## Problem 485

Let

\[A=\begin{bmatrix}

1 & -14 & 4 \\

-1 &6 &-2 \\

-2 & 24 & -7

\end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix}

4 \\

-1 \\

-7

\end{bmatrix}.\]
Find $A^{10}\mathbf{v}$.

You may use the following information without proving it.

The eigenvalues of $A$ are $-1, 0, 1$. The eigenspaces are given by

\[E_{-1}=\Span\left\{\, \begin{bmatrix}

3 \\

-1 \\

-5

\end{bmatrix} \,\right\}, \quad E_{0}=\Span\left\{\, \begin{bmatrix}

-2 \\

1 \\

4

\end{bmatrix} \,\right\}, \quad E_{1}=\Span\left\{\, \begin{bmatrix}

-4 \\

2 \\

7

\end{bmatrix} \,\right\}.\]

(*The Ohio State University, Linear Algebra Final Exam Problem*)