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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank Expansion of determinants, Solution of equation in the form of determinants and properties of determinants

  • question_answer
    Let \[\omega =-\frac{1}{2}+i\frac{\sqrt{3}}{2}\]. Then the value of the determinant \[\left| \,\begin{matrix}    1 & 1 & 1  \\    1 & -1-{{\omega }^{2}} & {{\omega }^{2}}  \\    1 & {{\omega }^{2}} & {{\omega }^{4}}  \\ \end{matrix}\, \right|\]is [IIT Screening 2002]

    A) \[3\omega \]

    B) \[3\omega (\omega -1)\]

    C) \[3{{\omega }^{2}}\]

    D) \[3\omega (1-\omega )\]

    Correct Answer: B

    Solution :

    \[\Delta =\left| \,\begin{matrix}    3 & 1 & 1  \\    0 & -1-{{\omega }^{2}} & {{\omega }^{2}}  \\    0 & {{\omega }^{2}} & \omega   \\ \end{matrix}\, \right|\] \[({{C}_{1}}\to {{C}_{1}}+{{C}_{2}}+{{C}_{3}})\] \[(\because \,\,1+\omega +{{\omega }^{2}}=0)\]  \[=3\,[\omega .\omega -{{\omega }^{4}}]=3({{\omega }^{2}}-\omega )\] \[=3\omega (\omega -1)\].


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