If $ω$ is a non-real cube root of unity, then $Δ=\begin{vmatrix}a_1 + b_1\,ω&a_1\,ω^2 + b_1&a_1 + b_1\,ω+c_1\,ω^2\\a_2 + b_2\,ω&a_2\,ω^2 + b_2&a_2 + b_2\,ω+c_2\,ω^2\\a_3 + b_3\,ω&a_3\,ω^2 + b_3&a_3 + b_3\,ω+c_3\,ω^2\end{vmatrix}$ is equal to

0

Applying $C_2 →C_2 (ω)$, we get

$Δ=\frac{1}{ω}\begin{vmatrix}a_1 + b_1\,ω&a_1 + b_1\,ω&a_1 + b_1\,ω+c_1\,ω^2\\a_2 + b_2\,ω&a_2 + b_2\,ω&a_2 + b_2\,ω+c_2\,ω^2\\a_3 + b_3\,ω&a_3 + b_3\,ω&a_3 + b_3\,ω+c_3\,ω^2\end{vmatrix}$

$⇒Δ=\frac{1}{ω}×0=0$  [∵ $C_1$ and $C_2$ are identical]