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

  • question_answer
    If A is a matrix such that \[{{A}^{2}}=A+I,\]where I is the unit matrix, then \[{{A}^{5}}\]is equal to

    A)  \[5A+I\]        

    B)  5A + 21

    C)  \[5A+3I\]       

    D)  \[5A+4I\]

    Correct Answer: C

    Solution :

    Given,   \[{{A}^{2}}=A+I\] Thus,   \[{{A}^{5}}={{A}^{2}}.\,{{A}^{2}}A\] \[=(A+I)\,(A+I)A\] \[=({{A}^{2}}+2AI+{{I}^{2}})A\] \[=(A+I+2A+I)A\] \[=(3A+2I)A\] \[=3{{A}^{2}}+2IA\] \[=3(A+I)+2IA\] \[=3A+3I+2A\] \[=5A+3I\]


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