JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

done Minors and Co-factors, Product of determinants Total Questions - 10

• question_answer1)
  The cofactor of the element '4' in the determinant \[\left| \,\begin{matrix}    1 & 3 & 5 & 1  \\    2 & 3 & 4 & 2  \\    8 & 0 & 1 & 1  \\    0 & 2 & 1 & 1  \\ \end{matrix}\, \right|\] is [MP PET 1987]
  A)
  4 done clear
  B)
  10 done clear
  C)
  - 10 done clear
  D)
  - 4 done clear
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• question_answer2)
  If \[\Delta =\left| \,\begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|\] and \[{{A}_{1}},{{B}_{1}},{{C}_{1}}\]denote the co-factors of \[{{a}_{1}},{{b}_{1}},{{c}_{1}}\] respectively, then the value of the determinant \[\left| \begin{matrix}    {{A}_{1}} & {{B}_{1}} & {{C}_{1}}  \\    {{A}_{2}} & {{B}_{2}} & {{C}_{2}}  \\    {{A}_{3}} & {{B}_{3}} & {{C}_{3}}  \\ \end{matrix} \right|\] is [MP PET 1989]
  A)
  \[\Delta \] done clear
  B)
  \[{{\Delta }^{2}}\] done clear
  C)
  \[{{\Delta }^{3}}\] done clear
  D)
  0 done clear
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• question_answer3)
  If in the determinant \[\Delta =\left| \begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\], \[{{A}_{1}},{{B}_{1}},{{C}_{1}}\] etc. be the co-factors of \[{{a}_{1}},{{b}_{1}},{{c}_{1}}\]etc., then which of the following relations is incorrect
  A)
  \[{{a}_{1}}{{A}_{1}}+{{b}_{1}}{{B}_{1}}+{{c}_{1}}{{C}_{1}}=\Delta \] done clear
  B)
  \[{{a}_{2}}{{A}_{2}}+{{b}_{2}}{{B}_{2}}+{{c}_{2}}{{C}_{2}}=\Delta \] done clear
  C)
  \[{{a}_{3}}{{A}_{3}}+{{b}_{3}}{{B}_{3}}+{{c}_{3}}{{C}_{3}}=\Delta \] done clear
  D)
  \[{{a}_{1}}{{A}_{2}}+{{b}_{1}}{{B}_{2}}+{{c}_{1}}{{C}_{2}}=\Delta \] done clear
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• question_answer4)
  If \[\omega \] is a cube root of unity and \[\Delta =\left| \begin{matrix}    1 & 2\omega   \\    \omega  & {{\omega }^{2}}  \\ \end{matrix} \right|\], then \[{{\Delta }^{2}}\]is equal to [RPET 1984]
  A)
  \[-\omega \] done clear
  B)
  \[\omega \] done clear
  C)
  1 done clear
  D)
  \[{{\omega }^{2}}\] done clear
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• question_answer5)
  If \[{{\Delta }_{1}}=\left| \,\begin{matrix}    1 & 0  \\    a & b  \\ \end{matrix}\, \right|\] and  \[{{\Delta }_{2}}=\left| \begin{matrix}    1 & 0  \\    c & d  \\ \end{matrix} \right|\], then \[{{\Delta }_{2}}{{\Delta }_{1}}\]is equal to [RPET 1984]
  A)
  ac done clear
  B)
  bd done clear
  C)
  \[(b-a)(d-c)\] done clear
  D)
  None of these done clear
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• question_answer6)
  If \[{{A}_{1}},{{B}_{1}},{{C}_{1}}\].... are respectively the co-factors of the elements \[{{a}_{1}},{{b}_{1}},{{c}_{1}}\],...... of the determinant \[\Delta =\left| \,\begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix}\, \right|\], then \[\left| \begin{matrix}    {{B}_{2}} & {{C}_{2}}  \\    {{B}_{3}} & {{C}_{3}}  \\ \end{matrix} \right|=\]
  A)
  \[{{a}_{1}}\Delta \] done clear
  B)
  \[{{a}_{1}}{{a}_{3}}\Delta \] done clear
  C)
  \[({{a}_{1}}+{{b}_{1}})\Delta \] done clear
  D)
  None of these done clear
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• question_answer7)
  Let \[A={{[{{a}_{ij}}]}_{n\times n}}\]be a square matrix and let \[{{c}_{ij}}\]be cofactor of \[{{a}_{ij}}\]in A. If \[C=[{{c}_{ij}}]\],then
  A)
  \[|C|\,=\,|A|\] done clear
  B)
    \[|C|\,=\,|A{{|}^{n-1}}\] done clear
  C)
  \[|C|\,=\,|A{{|}^{n-2}}\] done clear
  D)
  None of these done clear
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• question_answer8)
  \[\left| \,\begin{matrix}    {{\log }_{2}}512 & {{\log }_{4}}3  \\    {{\log }_{3}}8 & {{\log }_{4}}9  \\ \end{matrix}\, \right|\times \left| \,\begin{matrix}    {{\log }_{2}}3 & {{\log }_{8}}3  \\    {{\log }_{3}}4 & {{\log }_{3}}4  \\ \end{matrix}\, \right|\]= [Tamilnadu (Engg.) 2002]
  A)
  7 done clear
  B)
  10 done clear
  C)
  13 done clear
  D)
  17 done clear
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• question_answer9)
  If \[A=\left| \,\begin{matrix}    5 & 6 & 3  \\    -4 & 3 & 2  \\    -4 & -7 & 3  \\ \end{matrix}\, \right|\,,\]then cofactors of the elements of 2nd row are [RPET 2002]
  A)
  \[39,\,-3,\,11\] done clear
  B)
  \[-39,\,3,\,11\] done clear
  C)
  \[-39,\,27,\,11\] done clear
  D)
  \[-39,\,-3,\,11\] done clear
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• question_answer10)
  The minors of - 4 and 9 and the co-factors of - 4 and 9 in determinant \[\,\left| \,\begin{matrix}    -1 & -2 & 3  \\    -4 & -5 & -6  \\    -7 & 8 & 9  \\ \end{matrix}\, \right|\] are respectively [J & K 2005]
  A)
  42, 3; - 42, 3 done clear
  B)
  -42, -3; 42, -3 done clear
  C)
  42, 3; - 42, - 3 done clear
  D)
  42, 3; 42, 3 done clear
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