Let $ \alpha = e^{\frac{2\pi \iota}{5}}$ and the matrix $ M= \begin{pmatrix}1 & \alpha & \alpha^2 & \alpha^3 & \alpha^4\\ 0 & \alpha & \alpha^2 & \alpha^3 & \alpha^4\\ 0 & 0 & \alpha^2 & \alpha^3 & \alpha^4 \\ 0 & 0 & 0 & \alpha^3 & \alpha^4\\ 0 & 0 & 0 & 0 & \alpha^4 \end{pmatrix}$
Then the trace of the matrix $I + M + M^2$ is
- $-5$;
- $0$;
- $3$;
- $5$.
I am stuck on this problem. Can anyone help me please?
I got trace of the matrix $\operatorname{tr}(I+M+M^2) = 7 + \alpha + 2 \alpha^2 + \alpha^3 + 2 \alpha^4 + \alpha^6 +\alpha^8.$ Now what to do?