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If $n \ne 3k$ and 1, $\omega ,{\omega ^2}$ are the cube roots of unity, then $\Delta = \left| {\,\begin{array}{*{20}{c}}1&{{\omega ^n}}&{{\omega ^{2n}}}\\{{\omega ^{2n}}}&1&{{\omega ^n}}\\{{\omega ^n}}&{{\omega ^{2n}}}&1\end{array}\,} \right|$ has the value
$0$
$\omega $
${\omega ^2}$
$1$
Solution
(a) Applying ${C_1} \to {C_1} + {C_2} + {C_3}$, we get
$\Delta = \left| {\,\begin{array}{*{20}{c}}{1 + {\omega ^n} + {\omega ^{2n}}}&{{\omega ^n}}&{{\omega ^{2n}}}\\{1 + {\omega ^n} + {\omega ^{2n}}}&1&{{\omega ^n}}\\{1 + {\omega ^n} + {\omega ^{2n}}}&{{\omega ^{2n}}}&1\end{array}\,} \right| = \,\left| {\,\begin{array}{*{20}{c}}0&{{\omega ^n}}&{{\omega ^{2n}}}\\0&1&{{\omega ^n}}\\0&{{\omega ^{2n}}}&1\end{array}\,} \right|\, = \,\,0$,
$(\because \,{\text{ 1}} + {\omega ^n} + {\omega ^{2n}} = 0$, if n is not multiple of $3$ ).
