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(B) Given $w = \frac{-1-i \sqrt{3}}{2}$,which is the complex cube root of unity,denoted as $\omega$.We know that $1 + \omega + \omega^2 = 0$ and $\ome

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(B) Given $w = \frac{-1-i \sqrt{3}}{2}$,which is the complex cube root of unity,denoted as $\omega$.We know that $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.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 $1 + \omega + \omega^2 = 0$,we get:$\Delta = \left|\begin{array}{ccc}0 & \omega & \omega^2 \ 0 & \omega^2 & 1 \ 0 & 1 & \omega\end{array}ight|$Since all elements of the first column are $0$,the value of the determinant is $0$.

If $w = \frac{-1-i \sqrt{3}}{2}$ where $i = \sqrt{-1}$,then the value of $\left|\begin{array}{ccc}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{array}\right|$ is

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