If $\omega $ is a non-real cube root of unity and n is not a multiple of 3, then

$Δ=\begin{vmatrix}1 & ω^n & ω^{2n}\\ω^{2n} & 1 & ω^n\\ω^n & ω^{2n} & 1\end{vmatrix}$ is equal to

0

The correct answer is option : 0

Applying $C_1→C_1+C_2+C_3,$ we get

$Δ= (1 + ω^n +ω^{2n})\begin{vmatrix}1 & ω^n & ω^{2n}\\1 & 1 & ω^n\\1 & ω^{2n} & 1\end{vmatrix}$

$⇒Δ=0×\begin{vmatrix}1 & ω^n & ω^{2n}\\1 & 1 & ω^n\\1 & ω^{2n} & 1\end{vmatrix}$          $\begin{bmatrix}∵1+ω^n +ω^{2n}=0\\when\, n≠3k, k \in N \end{bmatrix}$

$⇒Δ=0$