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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
    \[\left| \,\begin{matrix}    bc & b{c}'+{b}'c & {b}'{c}'  \\    ca & c{a}'+{c}'a & {c}'{a}'  \\    ab & a{b}'+{a}'b & {a}'{b}'  \\ \end{matrix}\, \right|\] is equal to

    A) \[(ab-{a}'{b}')(bc-{b}'{c}')(ca-{c}'{a}')\]

    B) \[(ab+{a}'{b}')(bc+{b}'{c}')(ca+{c}'{a}')\]

    C) \[(a{b}'-{a}'b)(b{c}'-{b}'c)(c{a}'-{c}'a)\]

    D) \[(a{b}'+{a}'b)(b{c}'+{b}'c)(c{a}'+{c}'a)\]

    Correct Answer: C

    Solution :

    Trick: Put \[a=1,\,b=-1,\,c=0\] \[{a}'=2,\,{b}'=2,\,{c}'=1\] Then the determinant is \[\left| \,\begin{matrix}    0 & -1 & 2  \\    0 & 1 & 2  \\    -1 & 0 & 4  \\ \end{matrix}\, \right|=4\] Option (c) also gives the same value.


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