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J & K CET Engineering J and K - CET Engineering Solved Paper-2012

  • question_answer
    Matrix A is such that \[{{A}^{2}}=2A-I,\]where \[I\] is the identity matrix, then for \[n\ge 2,\,\,{{A}^{n}}\] is equal to

    A)  \[{{2}^{n-1}}A-(n-1)I\] 

    B)  \[{{2}^{n-1}}A-I\]

    C)  \[nA-(n-1)I\]   

    D)  \[nA-I\]

    Correct Answer: C

    Solution :

    Given, \[{{A}^{2}}=2A-I\] Now, \[{{A}^{3}}={{A}^{2}}.A\] \[=(2A-I)A\] \[=2{{A}^{2}}-A\] \[=2(2A-I)-A\] \[=3A-2I\] Now, \[{{A}^{4}}={{A}^{3}}.A\] \[=(3A-2I).A\] \[=3{{A}^{2}}-2A\] \[=3(2A-I)-2A\] \[=4A-3I\] Similarly,  \[{{A}^{n}}=nA-(n-1)I\]


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