If is an imaginary root of unity, then the value of | , * 20 c a&… - Vidyadip

MCQ

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

A
${a^3} + {b^3} + {c^3} - 3abc$
B
${a^2}b - {b^2}c$
✓
$0$
D
${a^2} + {b^2} + {c^2}$

✓

Answer

Correct option: C.

$0$

c
(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$.

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