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KVPY Sample Paper KVPY Stream-SX Model Paper-26

  • question_answer
    Let P be an \[m\times m\] matrix such that \[{{P}^{2}}=P.\] Then \[{{\left( I+P \right)}^{n}}\] equals (where I is identity matrix of order m)

    A) \[I+P\]

    B) \[I+nP\]

    C) \[I+{{2}^{n}}P\]

    D) \[I+({{2}^{n}}-1)P\]

    Correct Answer: D

    Solution :

    We have, \[{{P}^{2}}=P\] \[\Rightarrow \,\,\,\,{{P}^{-1}}{{P}^{2}}={{P}^{-1}}P\] \[\Rightarrow \,\,\,\,\,P=I\] \[\Rightarrow \,\,{{\left( I+P \right)}^{n}}={{\left( P+P \right)}^{n}}\]
    \[={{\left( 2P \right)}^{n}}={{2}^{n}}{{P}^{n}}={{2}^{n}}P\]
    \[=P+{{2}^{n}}\,\,\,P-P\]
    \[=I+\left( {{2}^{n}}-1 \right)p\]
     


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