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જો $\omega $ એકનું કાલ્પનિક ઘનમૂળ હોય , તો $\left| {\,\begin{array}{*{20}{c}}a&{b{\omega ^2}}&{a\omega }\\{b\omega }&c&{b{\omega ^2}}\\{c{\omega ^2}}&{a\omega }&c\end{array}\,} \right|$ મેળવો.
${a^3} + {b^3} + {c^3} - 3abc$
${a^2}b - {b^2}c$
$0$
${a^2} + {b^2} + {c^2}$
Solution
(c) We have $\left| {\,\begin{array}{*{20}{c}}a&{b{\omega ^2}}&{a\omega }\\{b\omega }&c&{b{\omega ^2}}\\{c{\omega ^2}}&{a\omega }&c\end{array}\,} \right|$
= $\left| {\,\begin{array}{*{20}{c}}{a(1 + \omega )}&{b{\omega ^2}}&{a\omega }\\{b(\omega + {\omega ^2})}&c&{b{\omega ^2}}\\{c({\omega ^2} + 1)}&{a\omega }&c\end{array}\,} \right|$ , $\{ {C_1} \to {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}\,} \right|$$ = {\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}\,} \right|$
= ${\omega ^2}\left| {\,\begin{array}{*{20}{c}}{ – a}&b&a\\{ – b}&c&b\\{ – c}&a&c\end{array}\,} \right|$= $ – {\omega ^2}\,\left| {\,\begin{array}{*{20}{c}}a&b&a\\b&c&b\\c&a&c\end{array}\,} \right|\, = 0$.
