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JEE Main & Advanced JEE Main Solved Paper-2016

  • question_answer
    If\[A=\left[ \begin{matrix}    5a & -b  \\    3 & 2  \\ \end{matrix} \right]\]and A adj A = A\[{{\text{A}}^{\text{T}}},\]then \[5a+b\]is equal to: [JEE Main Solved Paper-2016 ]

    A) 13                                          

    B) -1

    C) 5                                             

    D) 4

    Correct Answer: C

    Solution :

                    \[A=\left[ \begin{matrix}    5a & -b  \\    3 & 2  \\ \end{matrix} \right]\]and \[{{A}^{T}}=\left[ \begin{matrix}    5a & 3  \\    -b & 2  \\ \end{matrix} \right]\] \[A{{A}^{T}}=\left[ \begin{matrix}    25{{a}^{2}}+{{b}^{2}} & 15a-2b  \\    15a-2b & 13  \\ \end{matrix} \right]\] Now,\[AadjA=|A|{{I}_{2}}=\left[ \begin{matrix}    10a+3b & 0  \\    0 & 10a+3b  \\ \end{matrix} \right]\] Given \[A{{A}^{T}}=A.\,adj\,A\] 15a ?2b = 0                                                         ...(1) 10a + 3b = 13                                                      ...(2) Solving we get 5a = 2 and b = 3 \[\therefore \]5a + b = 5


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