The question is the following: 
The only way I can think of in doing this question would be for me to set variables for all values of A, and then using the given Eigenvectors and values to solve for the whole thing (every value in A).
The solution given here: Construct matrix given eigenvalues and eigenvectors is a little confusing, and I'm not entirely sure how to apply it.
If$$P=\begin{bmatrix}1&1&0\\0&1&1\\0&0&1\end{bmatrix}$$(the columns of $P$ are the entries of the three vectors), then$$P^{-1}AP=\begin{bmatrix}-1&0&0\\0&-3&0\\0&0&-4\end{bmatrix}$$and therefore$$P^{-1}A^5P=\begin{bmatrix}(-1)^5&0&0\\0&(-3)^5&0\\0&0&(-4)^5\end{bmatrix}.$$Can you take it from here?