Let $\alpha$ be a root of the equation $x^2 + x + 1 = 0$ and the matrix $A = \dfrac{1}{\sqrt{3}}\begin{pmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha^4 \end{pmatrix}$. Then the matrix $A^{31}$ is equal to: