Let α be a root of the equation x2+x+1=0 and the matrix  A=131111αα21α2α4 then the matrix A31 is equal to

# Let $\alpha$ be a root of the 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

1. A

$A$

2. B

${A}^{2}$

3. C

${A}^{3}$

4. D

$I$

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### Solution:

${x}^{2}+x+1=0\text{\hspace{0.17em}\hspace{0.17em}}⇒\text{\hspace{0.17em}\hspace{0.17em}}\alpha =\omega \text{\hspace{0.17em}}or\text{\hspace{0.17em}}{\omega }^{2}$

$A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& \omega & {\omega }^{2}\\ 1& {\omega }^{2}& \omega \end{array}\right]⇒{A}^{2}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]⇒{A}^{4}=I$

$\therefore {A}^{31}={A}^{28}.{A}^{3}=\left(I\right){A}^{3}={A}^{3}$

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