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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank Self Evaluation Test - Matrices

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

    [c] Given \[{{A}^{2}}=2A-I\] Now \[{{A}^{3}}=A({{A}^{2}})=A(2A-I)\] \[=2{{A}^{2}}-A=2(2A-I)-A=3A-2I\] \[{{A}^{4}}=A({{A}^{3}})=A(3A-2I)\] \[=3{{A}^{2}}-2A=3(2A-I)-2A=4A-3I\] Following this, we can say \[{{A}^{n}}=nA-(n-I)I\]


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