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KVPY Sample Paper KVPY Stream-SX Model Paper-15

  • question_answer
    A and B are square matrices and A is non-singular matrix, \[{{({{A}^{-1}}BA)}^{n}},\] \[n\in {{I}^{+}},\]is equal to -

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

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

    C) \[{{A}^{-1}}{{B}^{n}}A\]

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

    Correct Answer: C

    Solution :

    for \[n=2\] \[\Rightarrow ({{A}^{-1}}BA)\,\,({{A}^{-1}}BA)={{A}^{-1}}{{B}^{2}}A\] \[{{({{A}^{-1}}BA)}^{3}}={{({{A}^{-1}}BA)}^{2}}\,\,({{A}^{-1}}BA)=({{A}^{-1}}{{B}^{2}}A)\,\,({{A}^{-1}}BA)\]\[={{A}^{-1}}{{B}^{3}}A\]and so on thus \[{{({{A}^{-1}}BA)}^{n}}\] \[={{A}^{-1}}{{B}^{n}}A\]


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