Question

Find the value of $\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ 1& {\omega }^2& 1\\ {\omega }^2& \omega & 1\end{array}\right|$, where $\omega$ is complex cube root of unity.

Answer 1

$D=\omega \left|\begin{array}{ccc}1& 1& {\omega }^2\\ 1& \omega & 1\\ {\omega }^2& 1& 1\end{array}\right|$ $c_1\to c_1-c_2$

$\Rightarrow D=\omega \left|\begin{array}{ccc}0& 1& {\omega }^2\\ 1-\omega & \omega & 1\\ {\omega }^2-1& 1& 1\end{array}\right|$

Expanding along $R_1,D=\omega [-1(1-\omega -{\omega }^2+1)+{\omega }^2(1-\omega -{\omega }^3+\omega \left)\right]$

${\omega }^3=1$

$\Rightarrow D=\omega [-1+\omega +{\omega }^2-1+{\omega }^2-{\omega }^2-{\omega }^2+{\omega }^3]$

$\Rightarrow D=\omega [\omega -1]={\omega }^2-\omega$

As $\omega +{\omega }^2+1=0$

$\Rightarrow D=-1-\omega -\omega =-1-2\omega$.