If a_{r} = cos frac{2 r pi}{9} + i sin frac{2 | vedclass

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(D) Given $a_{r} = e^{i \frac{2 \pi r}{9}}$.Note that $a_{r} = (a_{1})^{r}$.The determinant is $\Delta = \left|\begin{array}{lll}a_{1} & a_{1}^{2} & a

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$0$

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(D) Given $a_{r} = e^{i \frac{2 \pi r}{9}}$.Note that $a_{r} = (a_{1})^{r}$.The determinant is $\Delta = \left|\begin{array}{lll}a_{1} & a_{1}^{2} & a_{1}^{3} \ a_{1}^{4} & a_{1}^{5} & a_{1}^{6} \ a_{1}^{7} & a_{1}^{8} & a_{1}^{9}\end{array}ight|$.Taking $a_{1}$ common from $C_{1}$,$a_{1}^{2}$ common from $C_{2}$,and $a_{1}^{3}$ common from $C_{3}$:$\Delta = a_{1} \cdot a_{1}^{2} \cdot a_{1}^{3} \left|\begin{array}{lll}1 & 1 & 1 \ a_{1}^{3} & a_{1}^{3} & a_{1}^{3} \ a_{1}^{6} & a_{1}^{6} & a_{1}^{6}\end{array}ight|$.Since all columns are identical,the value of the determinant is $0$.

If $a_{r} = \cos \frac{2 r \pi}{9} + i \sin \frac{2 r \pi}{9}$,$r = 1, 2, 3, \ldots$,$i = \sqrt{-1}$,then the determinant $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to:

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