A) \[0\]
B) \[-1\]
C) \[2\]
D) None of these
Correct Answer: A
Solution :
[a] \[\Delta =\left| \begin{matrix} {{a}_{1}}+{{b}_{1}}\omega & {{a}_{1}}{{\omega }^{2}}+{{b}_{1}} & {{c}_{1}}+{{b}_{1}}\overline{\omega } \\ {{a}_{2}}+{{b}_{2}}\omega & {{a}_{2}}{{\omega }^{2}}+{{b}_{2}} & {{c}_{2}}+{{b}_{2}}\overline{\omega } \\ {{a}_{3}}+{{b}_{3}}\omega & {{a}_{3}}{{\omega }^{2}}+{{b}_{3}} & {{c}_{3}}+{{b}_{3}}\overline{\omega } \\ \end{matrix} \right|\] Using \[{{C}_{2}}\to \omega {{C}_{2}}\] We have \[\Delta =\frac{1}{\omega }\left| \begin{matrix} {{a}_{1}}+{{b}_{1}}\omega & {{a}_{1}}{{\omega }^{3}}+{{b}_{1}}\omega & {{c}_{1}}+{{b}_{1}}\overline{\omega } \\ {{a}_{2}}+{{b}_{2}}\omega & {{a}_{2}}{{\omega }^{3}}+{{b}_{2}}\omega & {{c}_{2}}+{{b}_{2}}\overline{\omega } \\ {{a}_{3}}+{{b}_{3}}\omega & {{a}_{3}}{{\omega }^{2}}+{{b}_{3}}\omega & {{c}_{3}}+{{b}_{3}}\overline{\omega } \\ \end{matrix} \right|\] \[=\frac{1}{\omega }\left| \begin{matrix} {{a}_{1}}+{{b}_{1}}\omega & {{a}_{1}}+{{b}_{1}}\omega & {{c}_{1}}+{{b}_{1}}\overline{\omega } \\ {{a}_{2}}+{{b}_{2}}\omega & {{a}_{2}}+{{b}_{2}}\omega & {{c}_{2}}+{{b}_{2}}\overline{\omega } \\ {{a}_{3}}+{{b}_{3}}\omega & {{a}_{3}}+{{b}_{3}}\omega & {{c}_{3}}+{{b}_{3}}\overline{\omega } \\ \end{matrix} \right|=0\]
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