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(B) Let $\Delta = \left| \begin{array}{ccc} 1 & \omega & \omega^2 \ \omega & \omega^2 & 1 \ \omega^2 & 1 & \omega \end{array} ight|$.Applying the

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(B) Let $\Delta = \left| \begin{array}{ccc} 1 & \omega & \omega^2 \ \omega & \omega^2 & 1 \ \omega^2 & 1 & \omega \end{array} ight|$.Applying the column operation $C_1 	o C_1 + C_2 + C_3$:$\Delta = \left| \begin{array}{ccc} 1 + \omega + \omega^2 & \omega & \omega^2 \ \omega + \omega^2 + 1 & \omega^2 & 1 \ \omega^2 + 1 + \omega & 1 & \omega \end{array} ight|$.Since $1 + \omega + \omega^2 = 0$ for the cube roots of unity,we have:$\Delta = \left| \begin{array}{ccc} 0 & \omega & \omega^2 \ 0 & \omega^2 & 1 \ 0 & 1 & \omega \end{array} ight|$.Since all elements in the first column are $0$,the value of the determinant is $0$.

If $\omega$ is the cube root of unity,then $\left| \begin{array}{ccc} 1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega \end{array} \right| = $

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