JEE Main & Advanced JEE Main Solved Paper-2014

  • question_answer
    If A is an \[3\times 3\] non ? singular matrix such that AA′ = A′A and B = A−1 A′, then BB′ equals:   JEE Main  Solved  Paper-2014

    A) \[I+B\]                 

    B) \[I\]

    C) \[{{B}^{-1}}\]                                    

    D) \[({{B}^{-1}})'\]

    Correct Answer: B

    Solution :

    \[BBT=\left( {{A}^{-1}}{{A}^{T}} \right){{\left( {{A}^{-1}}{{A}^{T}} \right)}^{T}}\]\[={{A}^{-1}}.\left( {{A}^{T}}A \right){{\left( {{A}^{-1}} \right)}^{T}}\] \[=\left( {{A}^{-1}}A \right){{A}^{T}}.{{\left( {{A}^{-1}} \right)}^{T}}\] \[={{\left( {{A}^{-1}}A \right)}^{T}}=I\] \[=\left( {{A}^{-1}}{{A}^{T}} \right)\left( A\cdot {{\left( {{A}^{-1}} \right)}^{T}} \right)\] \[={{A}^{-1}}\left( A{{A}^{T}} \right){{\left( {{A}^{-1}} \right)}^{T}}\] \[={{A}^{T}}.{{\left( {{A}^{-1}} \right)}^{T}}\]

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