Finding a Higher Power of a Matrix given its Eigenvalues

Question

Question: Let $A$ be a $2\times 2$ matrix with Eigenvalues $\pm i$. Compute $A^{101}$.

I understand that $A^k = PD^kP^{-1}$, where in this case $D = \left[\begin{array}{cc} i & 0 \\ 0 & -i\end{array}\right]$. However, I am wondering how I can find the Eigenvectors to find $P$, given only the Eigenvalues? Or do I need the Eigenvectors at all?

Answer 1

We have $D^2=-I$ and so $A^2=-I$. Therefore, $A^{101}=A^{100}A=(-I)^{50}A=A$.