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If $\omega \ne 1$ is cube root of unity and $H = \left[ {\begin{array}{*{20}{c}}\omega &0\\0&\omega \end{array}} \right]$ then ${H^{70}}$ is equal to
$0$
$ - H$
$H$
${H^2}$
Solution
$H^{2}=\left[\begin{array}{ll}{\omega} & {0} \\ {0} & {\omega}\end{array}\right]\left[\begin{array}{ll}{\omega} & {0} \\ {0} & {\omega}\end{array}\right]=\left[\begin{array}{ll}{\omega^{2}} & {0} \\ {0} & {\omega^{2}}\end{array}\right]$
If $H^{k}=\left[\begin{array}{cc}{\omega^{k}} & {0} \\ {0} & {\omega}\end{array}\right] H^{k+1}=\left[\begin{array}{cc}{\omega^{k+1}} & {0} \\ {0} & {\omega^{k+1}}\end{array}\right]$
So by principle of mathematical induction, $H^{70}=\left[\begin{array}{cc}{\omega^{70}} & {0} \\ {0} & {\omega^{70}}\end{array}\right]=\left[\begin{array}{cc}{\omega^{69} \omega} & {0} \\ {0} & {\omega^{69} \omega}\end{array}\right]=\left[\begin{array}{cc}{\omega} & {0} \\ {0} & {\omega}\end{array}\right]=H$
