Let be a root of x^2+x+1=0, and the matrix A=1√3pmatrix1&1&11&&^21&^2&^4pmatrix.

Question

Let $\alpha$ be a root of $x^2+x+1=0$, and the matrix $A=\dfrac1{\sqrt3}\begin{pmatrix}1&1&1\1&\alpha&\alpha^2\1&\alpha^2&\alpha^4\end{pmatrix}$. Then the matrix $A^{31}$ is equal to:

Accepted Answer

The roots of $x^2+x+1=0$ are $\omega,\omega^2$; take $\alpha=\omega$, so $\alpha^4=\omega$.

Direct computation shows $A^2=\begin{pmatrix}1&0&0\0&0&1\0&1&0\end{pmatrix}$, and hence $A^4=I$.

$A^{31}=A^{28}\cdot A^3=(A^4)^7\cdot A^3=A^3$.

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