Let \( \alpha \) be a root of equation \( x^{2}+x+1=0 \) and the matrix \( A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1…

Let \( \alpha \) be a root of equation \( x^{2}+x+1=0 \) and the matrix \( A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & \alpha & \alpha^{2} \\ 1 & \alpha^{2} & \alpha^{4}\end{array}\right] \), then the matrix \( A^{31} \) is equal to :

[JEE Main 2020, 7 Jan (Shift 1)]