JEE Main & Advanced Sample Paper JEE Main Sample Paper-38

  • question_answer 72) Which of the following is correct?

    A)  If \[A\] and \[B\] are square matrices of order \[3\] such that \[|A|\,=-1,\,\,|B|\,=3\], then the determinant of \[3\,\,AB\] is equal to\[27\].

    B)  If \[A\] is an invertible matrix, then \[\det ({{A}^{-1}})\] is equal to\[\det (A)\]

    C)  If \[A\] and \[B\] are matrices of the same order, then\[{{(A+B)}^{2}}={{A}^{2}}+2AB+{{B}^{2}}\]is possible if\[AB=I\]

    D)  None of these

    Correct Answer: D

    Solution :

     [a] We have\[|AB|\,\,=\,\,|A||B|\] Also for a square matrix of order\[3,\,\,|kA|={{k}^{3}}|A|\] because each element of the matrix A is multiplied by \[k\] and hence in this case we will have k3 common \[\therefore \]\[|3AB|\,\,={{3}^{3}}|A||B|\,\,=27(-1)(3)=-81\] [b] Since \[A\] is invertible, therefore \[{{A}^{-1}}\] exists and             \[A{{A}^{-1}}=I=\det (A{{A}^{-1}})=\det (I)\] \[\Rightarrow \]            \[\det (A)\det ({{A}^{-1}})=1\] \[\Rightarrow \]            \[\det ({{A}^{-1}})=\frac{1}{\det (A)}\] [c]\[{{(A+B)}^{2}}=(A+B)(A+B)\] \[={{A}^{2}}+AB+BA+{{B}^{2}}\] \[={{A}^{2}}+2AB+{{B}^{2}}\]if\[AB=BA\].

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