left| begin{array}{ccc} 1 & 1 & 1 1 & omega^ | vedclass

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(A) Let $D = \left| \begin{array}{ccc} 1 & 1 & 1 \ 1 & \omega^2 & \omega \ 1 & \omega & \omega^2 \end{array} ight|$.Applying $C_1 	o C_1 - C_2$ a

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$3\sqrt{3}i$

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(A) Let $D = \left| \begin{array}{ccc} 1 & 1 & 1 \ 1 & \omega^2 & \omega \ 1 & \omega & \omega^2 \end{array} ight|$.Applying $C_1 	o C_1 - C_2$ and $C_2 	o C_2 - C_3$:$D = \left| \begin{array}{ccc} 0 & 0 & 1 \ 1-\omega^2 & \omega^2-\omega & \omega \ 1-\omega & \omega-\omega^2 & \omega^2 \end{array} ight|$.Expanding along the first row:$D = 1 \cdot [(1-\omega^2)(\omega-\omega^2) - (1-\omega)(\omega^2-\omega)]$$D = (\omega - \omega^2 - \omega^2 + \omega^4) - (\omega^2 - \omega - \omega^3 + \omega^2)$Since $\omega^3 = 1$ and $\omega^4 = \omega$:$D = (\omega - 2\omega^2 + \omega) - (2\omega^2 - \omega - 1)$$D = 2\omega - 2\omega^2 - 2\omega^2 + \omega + 1 = 3\omega - 4\omega^2 + 1$.Using $1 + \omega + \omega^2 = 0$,we have $1 = -\omega - \omega^2$:$D = 3\omega - 4\omega^2 - \omega - \omega^2 = 2\omega - 5\omega^2$.Alternatively,calculating directly:$D = 1(\omega^4 - \omega^2) - 1(\omega^2 - \omega) + 1(\omega - \omega^2) = \omega - \omega^2 - \omega^2 + \omega + \omega - \omega^2 = 3\omega - 3\omega^2 = 3(\omega - \omega^2)$.Substituting $\omega = \frac{-1 + i\sqrt{3}}{2}$ and $\omega^2 = \frac{-1 - i\sqrt{3}}{2}$:$D = 3 \left( \frac{-1 + i\sqrt{3}}{2} - \frac{-1 - i\sqrt{3}}{2} ight) = 3 \left( \frac{2i\sqrt{3}}{2} ight) = 3i\sqrt{3}$.

$\left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & \omega^2 & \omega \\ 1 & \omega & \omega^2 \end{array} \right| = $

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