If $\left(\omega \ne 1\right)$  is a cube root of unity, then $\left|\begin{array}{ccc}1& 1+\omega & 1+{\omega }^2\\ 1+{\omega }^2& 1& 1+\omega \\ 1+\omega & 1+{\omega }^2& 1\end{array}\right|$

$\left|\begin{array}{ccc}1& 1+\omega & 1+{\omega }^2\\ 1+{\omega }^2& 1& 1+\omega \\ 1+\omega & 1+{\omega }^2& 1\end{array}\right|$ 

Apply $c_1\to c_1+c_2+c_3$ ,

then determinant is

$\begin{array}{c}=\left|\begin{array}{ccc}2& 1+\omega & 1+{\omega }^2\\ 2& 1& 1+\omega \\ 2& 1+{\omega }^2& 1\end{array}\right|\\ R_2\to R_2-R_1 , R_3\to R_3-R_1\\ =2\left|\begin{array}{ccc}2& 1+\omega & 1+{\omega }^2\\ 0& -\omega & \omega -{\omega }^2\\ 0& -\omega +{\omega }^2& -{\omega }^2\end{array}\right|\end{array}$                 $\left(\sin ce 1+\omega +{\omega }^2=0\right)$

$=2\left({\omega }^3+{\omega }^2-{\omega }^3-{\omega }^3+{\omega }^4\right)$

$=2\left({\omega }^2-1+\omega \right)=-4$