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If $\omega$ is a complex cube root of unity and $A=\left[\begin{array}{cc}\omega & 0 \\ 0 & \omega\end{array}\right]$, then $A^{-1}=$
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$\mathrm{A}^{2}$
(C)
Given $A=\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right] \Rightarrow|A|=\left(\omega^{2}-0\right)=\omega^{2}$ and adj $A=\left[\begin{array}{cc}\omega & 0 \\ 0 & \omega\end{array}\right]$
$\therefore A^{-1}=\frac{1}{\omega^{2}}\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]=\left[\begin{array}{cc}\frac{1}{\omega} & 0 \\ 0 & \frac{1}{\omega}\end{array}\right]=\left[\begin{array}{cc}\omega^{2} & 0 \\ 0 & \omega^{2}\end{array}\right] \quad \ldots\left[\because \omega^{3}=1\right]$
$\therefore A^{2}=\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]=\left[\begin{array}{cc}\omega^{2} & 0 \\ 0 & \omega^{2}\end{array}\right]$
Given $A=\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right] \Rightarrow|A|=\left(\omega^{2}-0\right)=\omega^{2}$ and adj $A=\left[\begin{array}{cc}\omega & 0 \\ 0 & \omega\end{array}\right]$
$\therefore A^{-1}=\frac{1}{\omega^{2}}\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]=\left[\begin{array}{cc}\frac{1}{\omega} & 0 \\ 0 & \frac{1}{\omega}\end{array}\right]=\left[\begin{array}{cc}\omega^{2} & 0 \\ 0 & \omega^{2}\end{array}\right] \quad \ldots\left[\because \omega^{3}=1\right]$
$\therefore A^{2}=\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]=\left[\begin{array}{cc}\omega^{2} & 0 \\ 0 & \omega^{2}\end{array}\right]$
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