If the elements of matrix A are the reciprocals of elements of matrix $\left[\begin{array}{ccc}1 & \omega & \omega^{2} \ \omega & \omega^{2} & 1 \ \omega^{2} & 1 & \omega\end{array} ight]$, where $\omega$ is complex cube root of unity, then

Question

If the elements of matrix A are the reciprocals of elements of matrix $\left[\begin{array}{ccc}1 & \omega & \omega^{2} \ \omega & \omega^{2} & 1 \ \omega^{2} & 1 & \omega\end{array}ight]$, where $\omega$ is complex cube root of unity, then

Aicharya · Accepted Answer

$\begin{aligned} 	ext {We have, } \mathrm{A} &=\left[\begin{array}{ccc}1 & \frac{1}{\omega} & \frac{1}{\omega^{2}} \ \frac{1}{\omega} & \frac{1}{\omega^{2}} & 1 \ \frac{1}{\omega^{2}} & 1 & \frac{1}{\omega}\end{array}ight] \|\mathrm{A}| &=1\left\{\frac{1}{\omega^{3}}-1ight\}-\frac{1}{\omega}\left(\frac{1}{\omega^{2}}-\frac{1}{\omega^{2}}ight)+\frac{1}{\omega^{2}}\left(\frac{1}{\omega}-\frac{1}{\omega^{4}}ight) \ &=1(1-1)-0+\frac{1}{\omega^{2}}\left(\frac{1}{\omega}-\frac{1}{\omega}ight) \quad\left[\because \omega^{3}=1 	ext { and } \because \omega^{4}=\omegaight] \|\mathrm{A}| &=0 \Rightarrow \mathrm{A}^{-1} 	ext { does not exist } \end{aligned}$

If the elements of matrix A are the reciprocals of elements of matrix $\left[\begin{array}{ccc}1 & \omega &…

Solution