If is a complex cube root of unity, then the determinant | , * 20… - Vidyadip

MCQ

If $\omega $ is a complex cube root of unity, then the determinant $\left| {\,\begin{array}{*{20}{c}}2&{2\omega }&{ - {\omega ^2}}\\1&1&1\\1&{ - 1}&0\end{array}\,} \right| = $

✓
$0$
B
$1$
C
$-1$
D
None of these

✓

Answer

Correct option: A.

$0$

a
(a) $\Delta \equiv \left| {\,\begin{array}{*{20}{c}}2&{2\omega }&{ - {\omega ^2}}\\1&1&1\\1&{ - 1}&0\end{array}\,} \right| = \left| {\,\begin{array}{*{20}{c}}{2 + 2\omega + 2{\omega ^2}}&{2\omega }&{ - {\omega ^2}}\\{1 + 1 - 2}&1&1\\{1 - 1 - 0}&{ - 1}&0\end{array}\,} \right|$

$({C_1} \to {C_1} + {C_2} - 2{C_3})$

= $\left| {\,\begin{array}{*{20}{c}}0&{2\omega }&{ - {\omega ^2}}\\0&1&1\\0&{ - 1}&0\end{array}\,} \right|\, = \,0$.

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