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Question
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
Options
0
ω
ω2
1
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Solution
\[\Delta = \begin{vmatrix} 1 & w^n & w^{2n} \\ w^{2n} & 1 & w^n \\ w^n & w^{2n} & 1 \end{vmatrix}\]
\[ = \begin{vmatrix} 1 + w^n + w^{2n} & w^n & w^{2n} \\ w^{2n} + 1 + w^n & 1 & w^n \\ w^n + w^{2n} + 1 & w^{2n} & 1 \end{vmatrix} \left[\text{ Appplying }C_1 \to C_1 + C_2 + C_3 \right]\]
Now,
\[1 + w + w^2 = 0 \left[ \because\text{ w is a complex cube root of unity }\right]\]
\[ \Rightarrow 1 + w^n + w^{2n} = 0 \left[ \because\text{ n is not a multiple of 3 }\right]\]
\[ \Rightarrow \Delta = \begin{vmatrix} 0 & w^n & w^{2n} \\0 & 1 & w^n \\0 & w^{2n} & 1 \end{vmatrix} = 0 \]
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