A) \[3\omega \]
B) \[3\omega (\omega -1)\]
C) \[3{{\omega }^{2}}\]
D) \[3\omega (1-\omega )\]
Correct Answer: B
Solution :
Applying\[{{R}_{2}}\to {{R}_{2}}-{{R}_{1}};\,\,{{R}_{3}}\to {{R}_{3}}-{{R}_{1}}\], the given determinant becomes \[\Delta =\left| \begin{matrix} 1 & 1 & 1 \\ 0 & -2-{{\omega }^{2}} & {{\omega }^{2}}-1 \\ 0 & {{\omega }^{2}}-1 & \omega -1 \\ \end{matrix} \right|\] \[=1[(\omega -2)(-2-{{\omega }^{2}})-\{({{\omega }^{2}}-1)({{\omega }^{2}}-1)\}]\] \[-1(0)+1(0)\] \[=(-2-{{\omega }^{2}})(\omega -1)-{{({{\omega }^{2}}-1)}^{2}}\] \[=-2\omega +2-{{\omega }^{3}}+{{\omega }^{2}}-({{\omega }^{4}}+1-2{{\omega }^{2}})\] \[=-2\omega +1+{{\omega }^{2}}-(\omega +1-2{{\omega }^{2}})\] \[=3{{\omega }^{2}}-3\omega \] \[=3\omega (\omega -1)\]
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