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BCECE Engineering BCECE Engineering Solved Paper-2005

  • question_answer
    For any \[2\times 2\]matrix A, if \[A(adj\,A)=\left[ \begin{matrix}    10 & 0  \\    0 & 10  \\ \end{matrix} \right],\] then | A | i.e., det A is equal to:

    A)  20                         

    B)         100                       

    C)  10                         

    D)         0

    Correct Answer: C

    Solution :

    Since, \[A\,(adj\,A)=\left[ \begin{matrix}    10 & 0  \\    0 & 10  \\ \end{matrix} \right]\]                          ?(i) As we know, \[{{A}^{-1}}=\frac{1}{|A|}adj\,A\] \[\Rightarrow \]\[A\,{{A}^{-1}}|A|=A\,adj\,A\]                 \[=A(adj\,A)=|A|I\]              ?(ii) From Eqs. (i) and (ii), we get \[|A|I=10\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\]                 \[\Rightarrow \]    \[|A|I=10I\Rightarrow |A|=10\]


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