Question

Evaluate $\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& \omega & 1\end{array}\right|$ where $\omega$ is an imaginary cube root of unity.

Answer 1

$\to \left|\begin{array}{ccc}1& \omega & \omega \\ \omega & {\omega }^2& 1\\ {\omega }^2& \omega & 1\end{array}\right|$

$\to 1({\omega }^2-\omega )-\omega (\omega -{\omega }^2)+{\omega }^2({\omega }^2-{\omega }^4)$

$\to {\omega }^2-\omega -{\omega }^2+{\omega }^3+{\omega }^4-{\omega }^6$ ... (i)

$\omega$ = cube root of unity

${\omega }^3=1$ , $1+\omega +{\omega }^2=0$

$\to \therefore$ equation (i) becomes

$-\omega +1+\omega -1=0$

as ${\omega }^3=1,({\omega }^3{)}^2={\omega }^6=1$