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KVPY Sample Paper KVPY Stream-SX Model Paper-21

  • question_answer
    If the matrix \[\left[ \begin{matrix}    1 & 2a & 2  \\    0 & -1 & 1  \\    a & 1 & 3  \\ \end{matrix} \right]\] is not equivalent to \[{{I}_{3}}\] then a is, \[a\in R.\]

    A) 1

    B) 2

    C) 4

    D) No such 'a' exists

    Correct Answer: A

    Solution :

    \[A\,3\times 3\] matrix is equivalent to \[{{I}_{3}}\] if it is non-singular.
    Hence \[\left| \begin{matrix}    1 & 2a & 2  \\    0 & -1 & 1  \\    a & 1 & 3  \\ \end{matrix} \right|\,\,=0\]\[\Rightarrow \] \[-\,4+a\,(2a+2)=0\]\[\Rightarrow \]\[2{{a}^{2}}+2a-4=0\]\[\Rightarrow \]            \[(a+2)\,\,(a-1)=0\to a=-\,2,\,\,1\]


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