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

  • question_answer
    Let A, B, C, D be (not necessarily square) real matrices such that \[{{A}^{T}}=BCD;\text{ }{{B}^{T}}=CDA;\] \[{{C}^{T}}=DAB\] and \[{{D}^{T}}=ABC\] for the matrix \[S=ABCD,{{S}^{3}}=\]

    A) I

    B) \[{{S}^{2}}\]  

    C) S

    D) O

    Correct Answer: C

    Solution :

    [c] \[S=ABCD=A(BCD)=A{{A}^{T}}...(1)\] \[{{S}^{3}}=(ABCD)(ABCD)(ABCD)\] \[=(ABC)(DAB)(CDA)(BCD)\] \[={{D}^{T}}{{C}^{T}}{{B}^{T}}{{A}^{T}}\] \[={{(BCD)}^{T}}{{A}^{T}}=A{{A}^{T}}...(2)\] From (1) and (2), \[S={{S}^{3}}\]


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