Question
Let $$\omega $$ be a complex number such that $$2\omega + 1 = z$$ where $$z = \sqrt { - 3} .$$ If \[\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\
1\,\,\,\,\,\, - {\omega ^2} - 1\,\,\,\,\,\,\,\,\,{\omega ^2}\\
1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^2}\,\,\,\,\,\,\,\,\,\,\,{\omega ^7}
\end{array} \right| = 3k,\] then $$k$$ is equal to:
A.
1
B.
$$- z$$
C.
$$z$$
D.
$$- 1$$
Answer :
$$- z$$
Solution :
$$\eqalign{
& {\text{Given 2}}\omega {\text{ + 1}} = z; \cr
& z = \sqrt 3 i \cr
& \Rightarrow \,\,\omega = \frac{{\sqrt 3 i - 1}}{2} \cr} $$
⇒ $$\omega $$ is complex cube root of unity
Applying $${R_1} \to {R_1} + {R_2} + {R_3}$$
\[ = \left| {\begin{array}{*{20}{l}}
{3{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \,\,\,{\mkern 1mu} {\mkern 1mu} \,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \,\,\,\,\,\,\,\,\,\,\,\,\,\,0}\\
{1{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} - {\omega ^2} - 1{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \,\,{\omega ^2}}\\
{1{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \,\,\,{\mkern 1mu} \,\,\,\,{\omega ^2}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \,\,\,\,\,\,\,\,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \,{\mkern 1mu} {\mkern 1mu} {\omega }}
\end{array}} \right|\]
$$\eqalign{
& = 3\left( { - 1 - \omega - \omega } \right) \cr
& = - 3\left( {1 + 2\omega } \right) \cr
& = - 3z \cr
& \Rightarrow \,\,k = - z \cr} $$