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माना $\omega$ एक सम्मिश्र संख्या ऐसी है कि $2 w +1=z$ जहाँ $z=\sqrt{-3}$ है। यदि
$\left| {\begin{array}{*{20}{c}}1&1&1\\1&{ - {\omega ^2} - 1}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^7}}\end{array}} \right| = 3k$ है तो $k$ बराबर है:
$1$
$-z$
$z$
$-1$
Solution
Given $2\omega + 1 = z;$
$z = \sqrt {3i} $
$ \Rightarrow \omega = \frac{{\sqrt {3i} – 1}}{2}$
$ \Rightarrow \omega $ is complex cube root of unity
Applying ${R_1} \to {R_1} + {R_2} + {R_3}$
$ = \left| \begin{array}{l}
3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\
1\,\,\,\,\, – {\omega ^2} – 1\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^2}\\
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\omega
\end{array} \right|\,$
$ = 3\left( { – 1 – \omega – \omega } \right) = – 3\left( {1 + 2\omega } \right)\, = – 3z$
$ \Rightarrow k = – z$
