Let \(\omega=-\frac{1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}, \mathrm{i}=\sqrt{-1}\), then the value of \(\left|\begin{array}{ccc}1 & 1 & 1 \ 1 & -1-\omega^2 & \omega^2 \ 1 & \omega^2 & \omega^4\end{array}\right|\) is

\(\omega^2=\left(-\frac{1}{2}\right)^2+\left(\mathrm{i} \frac{\sqrt{3}}{2}\right)^2-2 \mathrm{i}\left(\frac{\sqrt{3}}{4}\right)=-\frac{1}{2}-\frac{\sqrt{3}}{2} \mathrm{i} \)
\( \omega^3=\frac{1}{4}+\frac{3}{4}=1 \)
\( \therefore 1+\omega+\omega^2=0\)
\(\therefore\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & -1-\omega^2 & \omega^2 \\ 1 & \omega^2 & \omega^4\end{array}\right|=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & \omega & \omega^2 \\ 1 & \omega^2 & \omega\end{array}\right|\)
\(=1\left(\omega^2-\omega^4\right)-1\left(\omega-\omega^2\right)+1\left(\omega^2-\omega\right) \)
\( =\omega^2-\omega^4-\omega+\omega^2+\omega^2-\omega \)
\( =-\omega^4+3 \omega^2-2 \omega \)
\( =-\omega+3 \omega^2-2 \omega \)
\( =3 \omega^2-3 \omega \)
\( =3 \omega(\omega-1)\)