Mathematics - Matrices Question with Solution | TestHub

n   = 1                                 n = 2
$\text{P}=\left[{\omega }^2\right]$$\text{P}=\left[\begin{array}{cc}{\omega }^2& {\omega }^3\\ {\omega }^3& {\omega }^4\end{array}\right]=\left[\begin{array}{cc}{\omega }^2& 1\\ 1& \omega \end{array}\right]$
${\text{P}}^2=\left[{\omega }^4\right]\ne 0$${\text{P}}^2=\left[\begin{array}{cc}{\omega }^4+1& \text{.......}\\ \text{.......}& \text{.......}\end{array}\right]\ne 0$
n = 3
$\text{P}=\left[\begin{array}{ccc}{\omega }^2& 1& \omega \\ 1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\end{array}\right] \left[\begin{array}{ccc}{\omega }^2& 1& \omega \\ 1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
Similarly${\text{P}}^2\ne 0$when n is not multiple of 3.