Step-by-step Solution:

We are given the determinant: \[ \left| \begin{array}{ccc} -\omega^2 & \omega^2 & \omega \\ \omega & \omega & -\omega^2 \\ -1 & \omega & -\omega^2 \end{array} \right| \] First, factor out \( \omega^2 \) from the first row: \[ = \omega^2 \left| \begin{array}{ccc} -\omega & \omega & -1 \\ -1 & 1 & -\omega \\ -1 & \omega & \omega^2 \end{array} \right| \] Next, simplify the determinant: \[ = \omega^2 \left| \begin{array}{ccc} 0 & \omega & -1 \\ 0 & 1 & -1 \\ \omega-1 & \omega & \omega^2 \end{array} \right| \] Finally, evaluating the determinant gives: \[ = -3 \omega^2 \]