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JEE Main & Advanced Sample Paper JEE Main Sample Paper-10

  • question_answer
    If A and B are square matrices of same order and A is non-singular then for a positive integer n, \[{{({{A}^{-1}}BA)}^{n}}\]equals

    A)  \[{{A}^{n}}{{B}^{n}}{{A}^{-n}}\]                              

    B)  \[{{A}^{-1}}{{B}^{n}}A\]

    C)  \[{{A}^{-n}}{{B}^{n}}{{A}^{n}}\]                              

    D)  None of these

    Correct Answer: B

    Solution :

     Let A and, Bare square matrices of same order and A is non-singular.                 For positive integer n, consider \[{{({{A}^{-1}}BA)}^{n}}=({{A}^{-1}}BA)({{A}^{-1}}BA)({{A}^{-1}}BA)\underbrace{.........}_{\text{n}\,\text{times}}({{A}^{-1}}BA)\]\[={{A}^{-1}}B(A{{A}^{-1}})B(A{{A}^{-1}})B(A{{A}^{-1}})...(A{{A}^{-1}})BA\] \[={{A}^{-1}}Bl\,Bl\,Bl...lBA\]      \[\because \{A{{A}^{-1}}=Bl\}\] \[={{A}^{-1}}\underbrace{B.B.B....}_{\text{n}\,\text{times}}BA={{A}^{-1}}{{B}^{n}}A\] \[\Rightarrow \]               \[{{({{A}^{-1}}BA)}^{n}}={{A}^{-1}}{{B}^{n}}A\]


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