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

W.
(a) \( \mathrm{A}^{3} \)
[JEE Main-2020 (January)]
(c) \( \mathrm{I}_{3} \)
(b) \( \mathrm{A}^{2} \)
(d) \( \mathrm{A} \)
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