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Question
If w is an imaginary cube root of unity, find the value of \[\begin{vmatrix}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{vmatrix}\]
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Solution
\[\begin{vmatrix}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{vmatrix}\]
\[ = \begin{vmatrix} 1 + w + w^2 & w & w^2 \\ w + w^2 + 1 & w^2 & 1\\ w^2 + 1 + w & 1 & w \end{vmatrix} \left[\text{ Applying }C_1 \to C_1 + C_2 + C_3 \right]\]
\[ = \begin{vmatrix} 0 & w 7 w^2 \\ 0 & w^2 & 1\\ 0 & 1 7 w \end{vmatrix} \left[ \because 1 + w + w^2 = 0,\text{ w is the imaginary cube root of unity }\right] \]
\[ = 0\]
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