Question
Let $$\omega = - \frac{1}{2} + i\frac{{\sqrt 3 }}{2},$$ then the value of the det.
\[\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\, - 1 - {\omega ^2}\,\,\,\,\,\,\,\,{\omega ^2}\,\,\\
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^4}
\end{array} \right|\] is
A.
$$3\omega $$
B.
$$3\omega \left( {\omega - 1} \right)$$
C.
$$3{\omega ^2}$$
D.
$$3\omega \left( {1 - \omega } \right)$$
Answer :
$$3\omega \left( {\omega - 1} \right)$$
Solution :
Operating $${R_1} + {R_2} + {R_3},$$ we get
\[\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\, - 1 - {\omega ^2}\,\,\,\,\,\,\,{\omega ^2}\,\,\\
1\,\,\,\,\,\,\,\,\,\,{\omega ^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^4}
\end{array} \right|\]
$$\eqalign{
& = 3\left[ { - \omega - 1 - \omega } \right] \cr
& = 3\left( {{\omega ^2} - \omega } \right) \cr} $$