If $ω$ 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 $ω$ $ω^2$ 1

0

Applying $C_1→C_1+C_2 + C_3$, we get $Δ=(1+ω^n+ω^{2n})\begin{vmatrix}1&ω^n&ω^{2n}\\1&1&ω^n\\1&ω^{2n}&1\end{vmatrix}$ $⇒Δ=0×\begin{vmatrix}1&ω^n&ω^{2n}\\1&1&ω^n\\1&ω^{2n}&1\end{vmatrix}$ $[∵1+ω^n+ω^{2n}=0\,when\, n≠3k,k∈N]$ $⇒Δ=0$