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KVPY Sample Paper KVPY Stream-SX Model Paper-4

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

    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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