If $\omega$ is a complex cube root of unity, then the matrix $A=\left[\begin{array}{ccc}1& {\omega }^2& \omega \\ {\omega }^2& \omega & 1\\ \omega & 1& {\omega }^2\end{array}\right]$ is a

We have 

$\left|A\right|=\left|\begin{array}{ccc}1& {\omega }^2& \omega \\ {\omega }^2& \omega & 1\\ \omega & 1& {\omega }^2\end{array}\right|=\left|\begin{array}{ccc}1+{\omega }^2+\omega & {\omega }^2& \omega \\ {\omega }^2+\omega +1& \omega & 1\\ \omega +1+{\omega }^2& 1& {\omega }^2\end{array}\right|=0$

using $C_1\to C_1+C_2+C_3$ and $1+\omega +{\omega }^2=0$

$\therefore$A is a singular matrix.