Evaluate :$\begin{vmatrix}1&\omega&\omega^2\\\omega&\omega^2&1\\\omega^2&1&\omega\end{vmatrix}$ ($\omega$: Cube root of unity)

Question

(A) 0 (B) 1 (C) 2 (D) 3

Answer 1

Apply $C_1\rightarrow C_1+C_2+C_3$

$\begin{vmatrix}1+\omega+\omega^2&\omega&\omega^2\\1+\omega+\omega^2&\omega^2&1\\1+\omega+\omega^2&1&\omega\end{vmatrix}$

$\Rightarrow \begin{vmatrix}0&\omega&\omega^2\\0&\omega^2&1\\0&1&\omega\end{vmatrix}$

$\Rightarrow 0$

Hence (a) is the correct answer.