Let α be a root of the equation x ^ 2 + x + 1 = 0 and the matrix A...

Here, $\alpha = \omega, {\omega}^{2}$ where $\omega$ is complex cube root of unity.

{A}^{2} = \dfrac{1}{3}\left[\begin{cases}1 & 1 & 1 \\ 1 & \omega & {\omega}^{2} \\ 1 & {\omega}^{2} & \omega\end{cases}\right]\left[\begin{cases}1 & 1 & 1 \\ 1 & \omega & {\omega}^{2} \\ 1 & {\omega}^{2} & \omega\end{cases}\right] = \left[\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}\right]

⇒ {A}^{4} = \left[\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}\right]\left[\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}\right] = \left[\begin{cases}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{cases}\right]

$⇒ {A}^{31} = {A}^{28} \times {A}^{3} = I \times {A}^{3} = {A}^{3}$