If \(\omega\) is a complex cube root of unity and \(A=\left[\begin{array}{cc}\omega & 0 \\ 0 & \omega\end{array}\right]\), then \(A^{-1}=\)

(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 \(A=\left[\begin{array}{cc}\omega & 0 \\ 0 & \omega\end{array}\right]\)
\(\therefore A^{-1}=\frac{1}{\omega^{2}}\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]=\left[\begin{array}{cc}\frac{1}{\omega} & 0 \\ 0 & \frac{1}{\omega}\end{array}\right]=\left[\begin{array}{cc}\omega^{2} & 0 \\ 0 & \omega^{2}\end{array}\right] \quad \ldots\left[\because \omega^{3}=1\right]\)
\(\therefore A^{2}=\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]\left[\begin{array}{ll}\omega & 0 \\ 0 & \omega\end{array}\right]=\left[\begin{array}{cc}\omega^{2} & 0 \\ 0 & \omega^{2}\end{array}\right]\)