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JEE Main & Advanced Sample Paper JEE Main Sample Paper-30

  • question_answer
    Let A be a square matrix of order 2 such that \[{{A}^{2}}-4A+4I=O\] where \[I\] is an identity matrix of order 2. If \[B={{A}^{5}}+4{{A}^{4}}+6{{A}^{3}}+4{{A}^{2}}+A\], then det.(B) is equal to

    A)  \[162\]                                    

    B)  \[{{(162)}^{2}}\]

    C)  \[256\]                        

    D)  \[{{(256)}^{2}}\]

    Correct Answer: B

    Solution :

    \[{{A}^{2}}-4A+4I=O\Rightarrow {{(A-2I)}^{2}}=O\Rightarrow A=2I\] \[B=A({{A}^{4}}+4{{A}^{3}}+6{{A}^{2}}+4A+I)=A{{(A+I)}^{4}}\] \[=2I\,{{(3I)}^{4}}=162\,I\] \[\therefore \det .(B)={{(162)}^{2}}\]


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