<s>[INST] Determinant of the matrix $(\omega^i \theta^j)_{i,j = 0,1,2}$ Given that $\omega=\frac{-1+\sqrt{-3}}{2}$  is a complex cube root of 1, $\theta$ is a real cube root of 2 and $\omega^{2}+\omega+1=0$, I am trying to maipulate the matrix
\begin{pmatrix}
1 & \theta &\theta^{2} \\
\omega & \omega \theta & \omega \theta^{2} \\
\omega^{2} & \omega^{2}\theta & \omega^{2}\theta^{2}
\end{pmatrix}
to end up at 
\begin{pmatrix}
1 & \theta &\theta^{2} \\
1 & \omega \theta & \omega^{2}\theta^{2}  \\
1 & \omega^{2}\theta & \omega \theta^{2}
\end{pmatrix}
However I am struggling to come up with a way of manipulating the matrix in order to end up here.
 [/INST] $$\begin{vmatrix}
1 & \theta &\theta^{2} \\
\omega & \omega \theta & \omega \theta^{2} \\
\omega^{2} & \omega^{2}\theta & \omega^{2}\theta^{2}
\end{vmatrix}
=\omega\,\omega^2\theta\,\theta^2\begin{vmatrix}
1 & 1&1\\
1&1& 1 \\
1 & 1 & 1
\end{vmatrix}
$$
The values of $\theta$ and $\omega$ don't matter.
</s>