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JEE Main & Advanced Mathematics Determinants & Matrices Question Bank Special types of matrices, Transpose, Adjoint and Inverse of matrices

  • question_answer
    If \[A=\left[ \begin{matrix}    1 & -1 & 1  \\    0 & 2 & -3  \\    2 & 1 & 0  \\ \end{matrix} \right]\] and \[B=(adj\,A)\], and \[C=5A,\] then \[\frac{|adjB|}{|C|}\]= [Kerala (Engg.) 2005]

    A) 5

    B) 25

    C) -1

    D) 1

    E) 125

    Correct Answer: D

    Solution :

    \[|A|=\left| \,\begin{matrix}    1 & -1 & 1  \\    0 & 2 & -3  \\    2 & 1 & 0  \\ \end{matrix}\, \right|\]\[=1[3]+1[6]+1[-4]\] \[=5\] B = adj\[\,A=\left[ \begin{matrix}    3 & 1 & 1  \\    -6 & -2 & 3  \\    -4 & -3 & 2  \\ \end{matrix} \right]\] adj \[B=\left[ \begin{matrix}    5 & -5 & 5  \\    0 & 10 & -15  \\    10 & 5 & 0  \\ \end{matrix} \right]=5A\]and  \[C=5A\]  C =adj B; \[|C|\] = |adj B|;  \ \[\frac{|adj\,B|}{|C|}\]=1.


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