Eigenvalus and trace of matrix

Question

Given matrix $A = \begin{bmatrix}1 & -2 & -1 \\ 1 & 2 & 1 \\ -1 & 0 & 1\end{bmatrix}$. I have to find its eigenvalues and trace of the matrix $A^{2014}$. I found engenvalues: $2$, $1 + i$, $1 - i$ but I don't know how to do the second task. Can you help me?

Answer 1

we will use the fact that if $\lambda$ is an eigenvalue of $A,$ then $\lambda^k$ is an eigenvalue of $A^k.$ it helps to write $1\pm i = \sqrt 2e^{\pm i\pi/4}$ so that $ (1\pm i)^{2014} = 2^{1007} e^{(i503\pi \pm \pi/2)}$.

so the eigenvalues of $A^{2014}$ are $\{2^{2014}, \pm 2^{1007}i \}$ and the trace of $A^{2014} = 2^{2014}$