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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 11

  • question_answer
    If A is a square matrix of order less than 4 such that \[|A-{{A}^{T}}|\ne 0\] and \[B=adj\,\,A,\] then \[adj\,({{B}^{2}}{{A}^{-1}}{{B}^{-1}}A)\] is

    A) \[A\]          

    B) \[B\]                     

    C) \[|A|A\]                

    D) \[|B|B\]

    Correct Answer: A

    Solution :

    [a] We know that \[A-{{A}^{T}}\] is skew symmetric matrix. But \[|A-{{A}^{T}}|\ne 0\] So, order of A is even. Thus, order of A is 2. Now, B = adj A \[\Rightarrow \,\,AB=BA=|A|\,I\] \[\Rightarrow \,\,{{B}^{-1}}{{A}^{-1}}={{A}^{-1}}{{B}^{-1}}\] \[\Rightarrow \,\,adj\,({{B}^{2}}{{A}^{-1}}{{B}^{-1}}A)=adj\,({{B}^{2}}{{B}^{-1}}{{A}^{-1}}A)\] \[=adj\,\,B=adj(adj\,\,A)=|A{{|}^{2-2}}A=A\] 


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