Question

If \(a_{r}=(\cos 2 r \pi+i \sin 2 r \pi)^{1 / 9}\), then the value of \(\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|\) is
(A) 1
(B) \(-1\)
(C) 0
(D) 2

Answer 1

Ans: (C)
Hint : Let \(\alpha=e^{i 2 \pi / 9}\) then given determinant \(=\left|\begin{array}{ccc}\alpha & \alpha^{2} & \alpha^{3} \\ \alpha^{4} & \alpha^{5} & \alpha^{6} \\ \alpha^{7} & \alpha^{8} & \alpha^{9}\end{array}\right|=\alpha^{12}\left|\begin{array}{ccc}1 & \alpha & \alpha^{2} \\ 1 & \alpha & \alpha^{2} \\ 1 & \alpha & \alpha^{2}\end{array}\right|=0\)

Answer 2

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Answer 3

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