Let $\mathrm{\alpha }$ be a root of the equation x² +x+ 1 =0 and the matrix $\mathrm{A}=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& \mathrm{\alpha }& {\mathrm{\alpha }}^2\\ 1& {\mathrm{\alpha }}^2& {\mathrm{\alpha }}^4\end{array}\right]$ then matrix A³¹ is

$\alpha$ is the root of $x^2+x+1=0\Rightarrow \alpha =\omega ,{\alpha }^2={\omega }^2,{\alpha }^4=\omega$
$\begin{array}{l}\therefore A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& \alpha & {\alpha }^2\\ 1& {\alpha }^2& {\alpha }^4\end{array}\right]=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right]\\ A^2=\frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\left[\begin{array}{ccc}1& 1& 1\\ 1& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right]=\frac{1}{3}\left[\begin{array}{lll}3& 0& 0\\ 0& 0& 3\\ 0& 3& 0\end{array}\right]=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]\right.\\ A^4=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]=\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=I.\therefore A^{31}={\left(A^4\right)}^7{A}^3=I^7{A}^3=A^3.\end{array}$