JEE Main & Advanced Sample Paper JEE Main Sample Paper-35

  • question_answer
    Let matrix A of order 3 is such that \[{{A}^{2}}=2A-I\] where \[I\] is an identity matrix of order 3. Then for \[n\in N\] and \[n\ge 2,\,{{A}^{n}}\] is equal to

    A)  \[nA-(n-l)I\]                     

    B)  \[nA-I\]

    C)  \[{{2}^{n-l}}A-(n-I)I\]                   

    D)  \[{{2}^{n-l}}A-I\]

    Correct Answer: A

    Solution :

    As we have \[{{A}^{2}}=2A-I\]             \[\Rightarrow \,\,{{A}^{2}}A\,=(2A-I)\,A=2{{A}^{2}}-IA\]             \[\Rightarrow \,{{A}^{3}}=2(2A-I)\,-IA=3A-2I\]             Similarly,                     \[{{A}^{4}}\,=4A-3I\]                                                 \[{{A}^{5}}=5A-4I\]                                                 ???????..                                                 ???????..                                                 ???????..           Hence,               \[{{A}^{n}}\,=nA\,-(n-1)I\]

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