Question

Prove that $\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& 1& \omega \end{array}\right|=0$

Answer 1

Solving given determinant, we get

$\left[1\right({\omega }^3-1)-\omega ({\omega }^2-{\omega }^2)+{\omega }^2(\omega -{\omega }^4\left)\right]$ ....... as $({\omega }^3=1)$

$⟹\left[\right(1-1)-\omega (0)+{\omega }^2(\omega -\omega \left)\right]$

$⟹0+0+0$

$⟹0$

Hence proved