Let omega be the complex number cos frac{2 pi | vedclass

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(A) Given $\omega = e^{i 2 \pi / 3}$,we know that $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.Let the determinant be $\Delta = \left|\begin{array}{

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(A) Given $\omega = e^{i 2 \pi / 3}$,we know that $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.Let the determinant be $\Delta = \left|\begin{array}{ccc} z+1 & \omega & \omega^2 \ \omega & z+\omega^2 & 1 \ \omega^2 & 1 & z+\omega \end{array}ight|$.Applying the operation $C_1 	o C_1 + C_2 + C_3$,we get:$\Delta = \left|\begin{array}{ccc} z+1+\omega+\omega^2 & \omega & \omega^2 \ z+\omega+\omega^2+1 & z+\omega^2 & 1 \ z+\omega^2+1+\omega & 1 & z+\omega \end{array}ight| = \left|\begin{array}{ccc} z & \omega & \omega^2 \ z & z+\omega^2 & 1 \ z & 1 & z+\omega \end{array}ight|$.Factoring out $z$ from the first column:$\Delta = z \left|\begin{array}{ccc} 1 & \omega & \omega^2 \ 1 & z+\omega^2 & 1 \ 1 & 1 & z+\omega \end{array}ight|$.Applying $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - R_1$:$\Delta = z \left|\begin{array}{ccc} 1 & \omega & \omega^2 \ 0 & z+\omega^2-\omega & 1-\omega^2 \ 0 & 1-\omega & z+\omega-\omega^2 \end{array}ight| = z[(z+\omega^2-\omega)(z+\omega-\omega^2) - (1-\omega^2)(1-\omega)]$.Expanding this,we find $\Delta = z(z^2) = z^3 = 0$.Thus,$z = 0$ is the only solution. The number of distinct complex numbers is $1$.

Let $\omega$ be the complex number $\cos \frac{2 \pi}{3} + i \sin \frac{2 \pi}{3}$. Then the number of distinct complex numbers $z$ satisfying $\left|\begin{array}{ccc} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{array}\right| = 0$ is equal to

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