If omega is an imaginary root of unity, then | vedclass

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(c) We have $\left| {\,\begin{array}{*{20}{c}}a&{b{\omega ^2}}&{a\omega }\{b\omega }&c&{b{\omega ^2}}\{c{\omega ^2}}&{a\omega }&

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(c) We have $\left| {\,\begin{array}{*{20}{c}}a&{b{\omega ^2}}&{a\omega }\{b\omega }&c&{b{\omega ^2}}\{c{\omega ^2}}&{a\omega }&c\end{array}\,} ight|$

= $\left| {\,\begin{array}{*{20}{c}}{a(1 + \omega )}&{b{\omega ^2}}&{a\omega }\{b(\omega + {\omega ^2})}&c&{b{\omega ^2}}\{c({\omega ^2} + 1)}&{a\omega }&c\end{array}\,} ight|$ , $\{ {C_1} 	o {C_1} + {C_3}\} $

= $\left| {\,\begin{array}{*{20}{c}}{ - a{\omega ^2}}&{b{\omega ^2}}&{a\omega }\{ - b}&c&{b{\omega ^2}}\{ - c\omega }&{a\omega }&c\end{array}\,} ight|$$ = {\omega ^2}\omega \,\,\left| {\,\begin{array}{*{20}{c}}{ - a}&b&{a{\omega ^2}}\{ - b}&c&{b{\omega ^2}}\{ - c}&a&{c{\omega ^2}}\end{array}\,} ight|$

= ${\omega ^2}\left| {\,\begin{array}{*{20}{c}}{ - a}&b&a\{ - b}&c&b\{ - c}&a&c\end{array}\,} ight|$= $ - {\omega ^2}\,\left| {\,\begin{array}{*{20}{c}}a&b&a\b&c&b\c&a&c\end{array}\,} ight|\, = 0$.

If $\omega $ is an imaginary root of unity, then the value of $\left| {\,\begin{array}{*{20}{c}}a&{b{\omega ^2}}&{a\omega }\\{b\omega }&c&{b{\omega ^2}}\\{c{\omega ^2}}&{a\omega }&c\end{array}\,} \right|$ is

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