If $\omega$ is a non-real cube root of unity and $n$ is not a multiple of 3, then
$\Delta=\left|\begin{array}{ccc}1 & \omega^{n} & \omega^{2 n} \\ \omega^{2 n} & 1 & \omega^{n} \\ \omega^{n} & \omega^{2 n} & 1\end{array}\right|$ is equal to
(a) 0
(b) $\omega$
(c) $\omega^{2}$
(d) 1
(a) 0
$\Delta=\mid 1 \quad w^{n} \quad w^{2 n}$
$w^{2 n} \quad 1 \quad w^{n}$
$w^{n} \quad w^{2 n} \quad 1 \mid$
$=\mid 1+w^{n}+w^{2 n} \quad w^{n} \quad w^{2 n}$
$w^{2 n}+1+w^{n} \quad 1 \quad w^{n}$
$w^{n}+w^{2 n}+1 \quad w^{2 n} \quad 1 \mid$ [Appplying $C_{1} \rightarrow C_{1}+C_{2}+C_{3}$ ]
Now,
$1+w+w^{2}=0$ $[\because w$ is a complex cube root of unity $]$
$\Rightarrow 1+w^{n}+w^{2 n}=0$ $[\because n$ is not a multiple of 3$]$
$\Rightarrow \Delta=\mid \begin{array}{lll}0 & w^{n} & w^{2 n}\end{array}$
$\begin{array}{lll}0 & 1 & w^{n}\end{array}$
$0 \quad w^{2 n} 1 \mid=0$
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