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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2013

  • question_answer
        If A is a non-singular\[3\times 3\]matrix such that\[A{{A}^{T}}={{A}^{T}}A\]and\[B={{A}^{-1}}{{A}^{T}}\]and\[|B|=1,\] then del\[(B-I)\]is equal to

    A)  1                                            

    B)  \[-1\]

    C)  0                                            

    D)  None of these

    Correct Answer: C

    Solution :

                    Given,       \[B={{A}^{-1}}.{{A}^{T}}\] \[\Rightarrow \]               \[{{B}^{T}}={{({{A}^{-1}}{{A}^{T}})}^{T}}=A.{{({{A}^{-1}})}^{T}}\] \[\Rightarrow \]               \[B.{{B}^{T}}={{A}^{-1}}{{A}^{T}}=A.{{({{A}^{-1}})}^{T}}\]                 \[={{A}^{-1}}.({{A}^{T}}A){{({{A}^{-1}})}^{T}}\]                 \[={{A}^{-1}}(A.{{A}^{T}}){{({{A}^{-1}})}^{T}}\]                 \[=({{A}^{-1}}A){{({{A}^{-1}}A)}^{T}}\]                 \[=I{{(I)}^{T}}=I\] Now, \[\det (B-I)=\det (B-B{{B}^{T}})\]                 \[=\det (B).\det (I-{{B}^{T}})\]                 \[=\det (B).\det (I-B)\]                 \[=1.\det (B-I)\]                 \[=\det (B-I)=0\]


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