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J & K CET Engineering J and K - CET Engineering Solved Paper-2006

  • question_answer
    If A is a non-singular matrix such that \[{{A}^{3}}=A+I,\] then the inverse of \[B={{A}^{6}}-{{A}^{5}}\]is

    A)  \[A\]            

    B)  \[{{A}^{-1}}\]

    C)  \[-A\]           

    D)  \[-{{A}^{-1}}\]

    Correct Answer: B

    Solution :

    Given,  \[B={{A}^{6}}-{{A}^{5}},\] where \[{{A}^{3}}=A+I\] \[\Rightarrow \] \[B={{({{A}^{3}})}^{2}}-{{A}^{3}}{{A}^{2}}\] \[={{(A+I)}^{2}}-(A+I){{A}^{2}}\] \[={{A}^{2}}+{{I}^{2}}+2AI-{{A}^{3}}-{{A}^{2}}I\] \[=I+2A-(A+I)\] \[\Rightarrow \] \[B=A\] \[\therefore \]  Inverse of \[B={{A}^{-1}}\]


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