JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

done Mock Test - Matrices Total Questions - 25

• question_answer1)

  Let A and B be two \[2\times 2\] matrices, Consider the statements

  (i) \[AB=O\Rightarrow A=O\] or \[B=0\]

  (ii) \[AB={{I}_{2}}\Rightarrow A={{B}^{-1}}\]

  (iii) \[{{(A+B)}^{2}}\]=\[{{A}^{2}}+2AB+{{B}^{2}}\]

  Then

  A)
   (i) and (ii) are false, (iii) is true done clear
  B)
   (ii) and (iii) are falsse, (i) is true done clear
  C)
   (i) is false, (ii) and (iii) are true done clear
  D)
   (i) and (iii) are false, (ii) is true done clear
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• question_answer2)
  If A and B are two matrices such that \[AB=B\]and \[BA=A,\]then
  A)
   \[{{({{A}^{5}}-{{B}^{5}})}^{3}}=A-B\] done clear
  B)
   \[{{({{A}^{5}}-{{B}^{5}})}^{3}}={{A}^{3}}-{{B}^{3}}\] done clear
  C)
   \[A-B\]is idempotent done clear
  D)
   \[A-B\]is nilpotent done clear
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• question_answer3)
    If A is a square matrix such that \[{{A}^{2}}=A\], then \[{{(I+A)}^{3}}-7A\] is
  A)
   3I                    done clear
  B)
   0 done clear
  C)
   I          done clear
  D)
   2I done clear
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• question_answer4)
  A is an involuntary matrix given by then the inverse of A/2 will be
  A)
   2A                   done clear
  B)
   \[\frac{{{A}^{-1}}}{2}\] done clear
  C)
   \[\frac{A}{2}\]  done clear
  D)
   \[{{A}^{2}}\] done clear
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• question_answer5)
  If both \[A-\frac{1}{2}I\] and \[A+\frac{1}{2}I\] are orthogonal matrices, then
  A)
   A is orthogonal done clear
  B)
   A is skew-symmetric of even order done clear
  C)
   \[{{A}^{2}}=\frac{3}{4}I\] done clear
  D)
   none of these done clear
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• question_answer6)

  If \[A=\left[ \begin{matrix}    a & b  \\    0 & a  \\ \end{matrix} \right]\] is nth root of \[{{I}_{2}}\], then choose the correct statements:

  (i) if n is odd, \[a=1,\text{ }b=0\]

  (ii) in n is odd, \[a=-1,\text{ }b=0\]

  (iii) if n is even, \[a=1,\text{ }b=0\]

  (iv) if n is even, \[a=-1,\text{ }b=0\]

   
  A)
   i, ii, iii done clear
  B)
  ii, iii, iv done clear
  C)
   i, ii, iii, iv          done clear
  D)
   i, iii, iv done clear
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• question_answer7)
  Let A be an nth-order square matrix and B be its adjoint, then \[\left| AB+K{{I}_{n}} \right|\]is (where K is a scalar quantity)
  A)
   \[{{(\left| A \right|+K)}^{n-2}}\]          done clear
  B)
   \[{{(\left| A \right|+K)}^{n}}\] done clear
  C)
   \[{{(\left| A \right|+K)}^{n-1}}\]          done clear
  D)
   none of these done clear
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• question_answer8)
  If and if A is invertible, then which of the following is not true?
  A)
   \[\left| A \right|=\left| B \right|\] done clear
  B)
   \[\left| A \right|=-\left| B \right|\] done clear
  C)
   \[\left| adj\,A \right|=\left| adj\,B \right|\] done clear
  D)
   A is invertible if and only if B is invertible done clear
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• question_answer9)
  If A and B are two nonsingular matrices of the same order such that B?=I, for some positive integer r>1, then \[{{A}^{-1}}{{B}^{\,r-1}}A-{{A}^{-1}}{{B}^{-1}}A\]=
  A)
  I                     done clear
  B)
  2I done clear
  C)
  O             done clear
  D)
  -I done clear
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• question_answer10)
  If \[A=\left[ \begin{matrix}    1 & \tan x  \\    -\tan x & 1  \\ \end{matrix} \right]\], then \[{{A}^{T}}{{A}^{-1}}\]is
  A)
   \[\left[ \begin{matrix}    -\cos 2x & \sin 2x  \\    -\sin 2x & \cos 2x  \\ \end{matrix} \right]\] done clear
  B)
   \[\left[ \begin{matrix}    \cos 2x & -\sin 2x  \\    \sin 2x & \cos 2x  \\ \end{matrix} \right]\] done clear
  C)
   \[\left[ \begin{matrix}    \cos 2x & \cos 2x  \\    \sin 2x & \sin 2x  \\ \end{matrix} \right]\] done clear
  D)
   none of these done clear
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• question_answer11)
    If  then \[A{{(\alpha ,\beta )}^{-1}}\]is equal to
  A)
   \[A(-\alpha ,-\beta )\]      done clear
  B)
   \[A(-\alpha ,\beta )\] done clear
  C)
   \[A(\alpha ,-\beta )\]       done clear
  D)
   \[A(\alpha ,\beta )\] done clear
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• question_answer12)
  If \[{{A}^{3}}=0\], then I=A+\[{{A}^{2}}\]equals
  A)
  \[I-A\]   done clear
  B)
  \[{{(I+{{A}^{1}})}^{-1}}\] done clear
  C)
  \[{{(I-A)}^{-1}}\]        done clear
  D)
  none of these done clear
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• question_answer13)
  If A and B are square matrices of the same order and A is nonsingular, then for a positive integer n, \[{{({{A}^{-1}}BA)}^{n}}\] is equal
  A)
   \[{{A}^{-n}}{{B}^{n}}{{A}^{n}}\] done clear
  B)
   \[{{A}^{n}}{{B}^{n}}{{A}^{-n}}\] done clear
  C)
   \[{{A}^{-1}}{{B}^{n}}A\]    done clear
  D)
   \[n({{A}^{-1}}BA)\] done clear
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• question_answer14)
  \[{{(-A)}^{-1}}\] is always equal to (where A is nth-order square matrix)
  A)
   \[{{(-1)}^{n}}{{A}^{-1}}\] done clear
  B)
   \[-{{A}^{-1}}\] done clear
  C)
   \[{{(-1)}^{n-1}}{{A}^{-1}}\] done clear
  D)
   none of these done clear
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• question_answer15)
  If \[\left[ \begin{matrix}    2 & 1  \\    3 & 2  \\ \end{matrix} \right]A\left[ \begin{matrix}    -3 & 2  \\    5 & -3  \\ \end{matrix} \right]=\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\], then A=
  A)
  \[\left[ \begin{matrix}    1 & 1  \\    1 & 0  \\ \end{matrix} \right]\]       done clear
  B)
  \[\left[ \begin{matrix}    1 & 1  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 0  \\    1 & 1  \\ \end{matrix} \right]\]       done clear
  D)
  \[-\left[ \begin{matrix}  1 & 1  \\    1 & 0  \\ \end{matrix} \right]\] done clear
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• question_answer16)
  If A is symmetric as well as skew-symmetric matrix, then A is
  A)
   diagonal matrix done clear
  B)
   null matrix done clear
  C)
   triangular matrix done clear
  D)
   none of these done clear
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• question_answer17)
  The number of solutions of the matrix equation\[{{X}^{2}}=\left[ \begin{matrix}    1 & 1  \\    2 & 3  \\ \end{matrix} \right]\] is
  A)
   more than 2      done clear
  B)
   2 done clear
  C)
   0         done clear
  D)
   1 done clear
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• question_answer18)
  Given that matrix . If \[xyz=60\] and \[8x+4y+3z=20,\] then A (adjA) is equal to
  A)
              done clear
  B)
  done clear
  C)
  done clear
  D)
  done clear
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• question_answer19)
  Elements of a matrix A of order \[10\times 10\] are defined is \[{{a}_{ij}}={{w}^{i+j}}\] (where w is cube root of unity), then tr of the matrix is
  A)
   0                     done clear
  B)
   1 done clear
  C)
   3                     done clear
  D)
   none of these done clear
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• question_answer20)
  If \[A=\left[ \begin{matrix}    a & b  \\    b & a  \\ \end{matrix} \right]\]and \[{{A}^{2}}=\left[ \begin{matrix}    \alpha  & \beta   \\    \beta  & \alpha   \\ \end{matrix} \right]\], then
  A)
  \[\alpha ={{a}^{2}}+{{b}^{2}},\beta =ab\] done clear
  B)
  \[\alpha ={{a}^{2}}+{{b}^{2}},\beta =2ab\] done clear
  C)
  \[\alpha ={{a}^{2}}+{{b}^{2}},\beta ={{a}^{2}}-{{b}^{2}}\] done clear
  D)
  \[\alpha =2ab,\,\,\beta ={{a}^{2}}+{{b}^{2}}\] done clear
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• question_answer21)
  Let \[A=\left[ \begin{matrix}    0 & \alpha   \\    0 & 0  \\ \end{matrix} \right]\] and \[{{(A+1)}^{50}}-50A=\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right]\].Then the value of a+b+c+d is _____.
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• question_answer22)
  If matrix A is given by \[A=\left[ \begin{matrix}    6 & 11  \\    2 & 4  \\ \end{matrix} \right]\], then the determinant of \[{{A}^{2005}}-6{{A}^{2004}}\] is _______.
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• question_answer23)
  If \[A={{[{{a}_{i\,j}}]}_{4\,\times \,4}},\] such that \[{{a}_{ij}}=\left\{ \begin{matrix}    2,\,\,when\,\,\,i=j  \\    0,\,\,when\,\,\,i\ne j  \\ \end{matrix} \right.\] then \[\left\{ \frac{\det (adj(adj\,A))}{7} \right\}\] is (where {.} represents fractional part function) ______.
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• question_answer24)
  If A is nonsingular and \[(A-2I)\]\[(A-4I)=O\], then \[\frac{1}{6}A+\frac{4}{3}{{A}^{-1}}\] is equal to KI. Then the value of k is _______.
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• question_answer25)
  Let  and . If B is the inverse of A, then \[\alpha \] is _____.
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