## Given All Eigenvalues and Eigenspaces, Compute a Matrix Product

## Problem 189

Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces

\[E_2=\Span\left \{\quad \begin{bmatrix}

1 \\

1 \\

1 \\

1

\end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix}

1 \\

2 \\

1 \\

1

\end{bmatrix},\quad \begin{bmatrix}

1 \\

1 \\

1 \\

2

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

Calculate $C^4 \mathbf{u}$ for $\mathbf{u}=\begin{bmatrix}

6 \\

8 \\

6 \\

9

\end{bmatrix}$ if possible. Explain why if it is not possible!

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

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