If $\omega$is imaginary root of unity, then the value of $\left| \begin{matrix} a & b\omega^{2} & a\omega \ b\omega & c & b\omega^{2} \ c\omega^{2} & a\omega & c \end{matrix} ight|$ is

Question

If $\omega$is imaginary root of unity, then the value of $\left| \begin{matrix} a & b\omega^{2} & a\omega \ b\omega & c & b\omega^{2} \ c\omega^{2} & a\omega & c \end{matrix} ight|$ is

Dr. Abubacker · Accepted Answer

(c)
Let $\mathrm{\Delta} = \left| \begin{matrix}
a & b\omega^{2} & a\omega \
b\omega & c & b\omega^{2} \
c\omega^{2} & a\omega & c
\end{matrix} ight|$
Applying $C_{3} ightarrow C_{3} - \omega C_{1}$
=$\left| \begin{matrix}
a & b\omega^{2} & 0 \
b\omega & c & 0 \
c\omega^{2} & a\omega & 0
\end{matrix} ight|$
$= 0$

Q. If $\omega$is imaginary root of unity, then the value of $\left| \begin{matrix} a & b\omega^{2} & a\omega \\ b\omega & c & b\omega^{2} \\ c\omega^{2} & a\omega & c \end{matrix} \right|$ is

Explanation

Reviewed by Dr. Abubacker

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