A) 0
B) \[3\sqrt{3}i\]
C) \[-3\sqrt{3}i\]
D) \[1\]
Correct Answer: C
Solution :
\[\left| \begin{matrix} 1 & 1 & 1 \\ 1 & \omega & {{\omega }^{2}} \\ 1 & {{\omega }^{2}} & \omega \\ \end{matrix} \right|\] \[=\left| \begin{matrix} 1+1+1 & 1 & 1 \\ 1+\omega +{{\omega }^{2}} & \omega & {{\omega }^{2}} \\ 1+{{\omega }^{2}}+\omega & {{\omega }^{2}} & \omega \\ \end{matrix} \right|\] \[[{{C}_{1}}\to {{C}_{1}}+{{C}_{2}}+{{C}_{3}}]\] \[=\left| \begin{matrix} 3 & 1 & 1 \\ 0 & \omega & {{\omega }^{2}} \\ 0 & {{\omega }^{2}} & \omega \\ \end{matrix} \right|\]\[[\because 1+\omega +{{\omega }^{2}}=0]\] \[=3[{{\omega }^{2}}-{{\omega }^{4}}]\] \[=3[{{\omega }^{2}}-{{\omega }^{3}}\omega ]\] \[=3[{{\omega }^{2}}-\omega ]\] \[[\because {{\omega }^{3}}=1]\] \[=3\left[ -\frac{1}{2}-\frac{\sqrt{3}}{2}i+\frac{1}{2}-\frac{\sqrt{3}}{2}i \right]\] \[=3[-\sqrt{3}i]\] \[=-3\sqrt{3}i\]
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