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STD 12 - 3 and 4 . determinant and metrices — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 3 and 4 . determinant and metrices4 Marks
MCQ
Let $\omega = - \frac{1}{2} + i\frac{{\sqrt 3 }}{2}$. Then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{ - 1 - {\omega ^2}}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^4}}\end{array}\,} \right|$ is
  • A
    $3\omega $
  • ✓
    $3\omega (\omega - 1)$
  • C
    $3{\omega ^2}$
  • D
    $3\omega (1 - \omega )$
✓

Answer

Correct option: B.
$3\omega (\omega - 1)$
b
(b) $\Delta = \left| {\,\begin{array}{*{20}{c}}3&1&1\\0&{ - 1 - {\omega ^2}}&{{\omega ^2}}\\0&{{\omega ^2}}&\omega \end{array}\,} \right|$

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

$(\because\,\,1 + \omega  + {\omega ^2} = 0)$

$ = 3\,[\omega .\omega - {\omega ^4}] = 3({\omega ^2} - \omega )$ $ = 3\omega (\omega - 1)$.

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