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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
    If \[\left| \,\begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|=5\]; then the value of  \[\left| \,\begin{matrix}    {{b}_{2}}{{c}_{3}}-{{b}_{3}}{{c}_{2}} & {{c}_{2}}{{a}_{3}}-{{c}_{3}}{{a}_{2}} & {{a}_{2}}{{b}_{3}}-{{a}_{3}}{{b}_{2}}  \\    {{b}_{3}}{{c}_{1}}-{{b}_{1}}{{c}_{3}} & {{c}_{3}}{{a}_{1}}-{{c}_{1}}{{a}_{3}} & {{a}_{3}}{{b}_{1}}-{{a}_{1}}{{b}_{3}}  \\    {{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}} & {{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}} & {{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}  \\ \end{matrix}\, \right|\] is [Tamilnadu (Engg.) 2002]

    A) 5

    B) 25

    C) 125

    D) 0

    Correct Answer: B

    Solution :

    Required determinant \[|adj\,A|\]=\[|A{{|}^{3-1}}\],\[\,\text{where}A=\left| \,\begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|\]             \[={{5}^{2}}=25,\]  \[(\because \,\,|adj\,A|=|a{{|}^{n-1}})\]


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