This one is going to take a lot of calculation. You might consider checking your work on something like Wolfram|Alpha. Only to check your work! You're on your honor. (Hey I managed it so you can too.)

In any way you feel like, find the matrix, A, that generates the following eigenvectors and their associated eigenvalues.

$\displaystyle \lambda _1 = 6 \text{ and } | v_1 > = \left ( \begin{matrix} 2 \\ 0 \\ 1 \end{matrix} \right )$

$\displaystyle \lambda _2 = 0 \text{ and } | v_2 > = \left ( \begin{matrix} 1 \\ 1 \\ 0 \end{matrix} \right )$

$\displaystyle \lambda _3 = -6 \text{ and } | v_1 > = \left ( \begin{matrix} -1 \\ 1 \\ 2 \end{matrix} \right )$

Find a formula for $\displaystyle A^n$. and their eigenvectors and eigenvalues. (Hint: Consider even n and odd n. They have different formulas.)

-Dan