JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

done Self Evaluation Test - Matrices Total Questions - 60

• question_answer1)
  The matrix \[A=\left[ \begin{matrix}    -5 & -8 & 0  \\    3 & 5 & 0  \\    1 & 2 & -1  \\ \end{matrix} \right]\] is
  A)
  Idempotent matrix done clear
  B)
  Involutory matrix done clear
  C)
  Nilpotent matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  If \[{{A}^{k}}=0\] (A is nilpotent with index k),\[{{(I-A)}^{p}}=I+A+{{A}^{2}}+....+{{A}^{k-1}},\] thus p is,
  A)
  -1 done clear
  B)
  -2 done clear
  C)
  ½ done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  If C is skew-symmetric matrix of order n and X is \[n\times 1\] column matrix, then X'CX is a
  A)
  Scalar matrix done clear
  B)
  Unit matrix done clear
  C)
  Null matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer4)
  The matrix \[A=\left[ \begin{matrix}    1 & 3 & 2  \\    1 & x-1 & 1  \\    2 & 7 & x-3  \\ \end{matrix} \right]\] will have inverse for every real number x except for
  A)
  \[x=\frac{11\pm \sqrt{5}}{2}\] done clear
  B)
  \[x=\frac{9\pm \sqrt{5}}{2}\] done clear
  C)
  \[x=\frac{11\pm \sqrt{3}}{2}\] done clear
  D)
  \[x=\frac{9\pm \sqrt{3}}{2}\] done clear
  View Solution play_arrow

• question_answer5)
  If A and B are two matrices such that AB = A and BA=B, then which one of the following is correct?
  A)
  \[{{({{A}^{T}})}^{2}}={{A}^{T}}\] done clear
  B)
  \[{{({{A}^{T}})}^{2}}={{B}^{T}}\] done clear
  C)
  \[{{({{A}^{T}})}^{2}}={{({{A}^{-1}})}^{-1}}\] done clear
  D)
  None of the above done clear
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• question_answer6)
  If \[{{A}_{1}},{{A}_{3}},.......,{{A}_{2n-1}}\] are n skew-symmetric matrices of same order, then \[B=\sum\limits_{r=1}^{n}{(2r-1){{({{A}_{2r-1}})}^{2r-1}}}\] will be
  A)
  Symmetric done clear
  B)
  Skew-symmetric done clear
  C)
  Neither symmetric nor skew-symmetric done clear
  D)
  Data is adequate done clear
  View Solution play_arrow

• question_answer7)
  If \[A=\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 0  \\    a & b & -1  \\ \end{matrix} \right]\] and I is the unit matrix of order 3, then \[{{A}^{2}}+2{{A}^{4}}+4{{A}^{6}}\] is equal to
  A)
  \[7{{A}^{8}}\] done clear
  B)
  \[7{{A}^{7}}\] done clear
  C)
  8I done clear
  D)
  6I done clear
  View Solution play_arrow

• question_answer8)
  The number of all possible matrices of order \[3\times 3\]with each entry 0 or 1 is
  A)
  18 done clear
  B)
  512 done clear
  C)
  81 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  If number of elements is 20 then how many different types of matrices can be formed if number of rows is always even?
  A)
  3 done clear
  B)
  4 done clear
  C)
  5 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer10)
  sLet \[A=\left[ \begin{matrix}    1 & 2  \\    3 & 4  \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    a & 0  \\    0 & b  \\ \end{matrix} \right]\] where a, b are natural numbers, then which one of the following is correct?
  A)
  There exist more than one but finite number of B's such that AB = BA done clear
  B)
  There exists exactly one B such that AB = BA done clear
  C)
  There exist infinitely many B's such that AB=BA done clear
  D)
  There cannot exist any B such that AB = BA done clear
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• question_answer11)
  Elements of a matrix A of order \[10\times 10\] are defined as \[{{a}_{ij}}={{w}^{i+j}}\] (where w is cube root of unity), then trof the matrix is
  A)
  0 done clear
  B)
  1 done clear
  C)
  3 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  If B, C are square matrices of order n and if \[A=B+C,\text{ }BC=CB,\text{ }{{C}^{2}}=0\], then for any positive integer \[N,{{A}^{N+1}}={{B}^{K}}[B+(N+1)C],\]  then K/N is
  A)
   1 done clear
  B)
  ½ done clear
  C)
  2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  If \[A=\left[ \begin{matrix}    0 & -1  \\    1 & 0  \\ \end{matrix} \right]\], then \[{{A}^{16}}\] is equal to:
  A)
  \[\left[ \begin{matrix}    0 & -1  \\    1 & 0  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    0 & 1  \\    1 & 0  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -1 & 0  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer14)
  Let \[A=\left[ \begin{matrix}    x+y & y  \\    2x & x-y  \\ \end{matrix} \right],B=\left[ \begin{matrix}    2  \\    -1  \\ \end{matrix} \right]\] and \[C=\left[ \begin{matrix}    3  \\    2  \\ \end{matrix} \right]\] If \[AB=C,\] then what is \[{{A}^{2}}\] equal to?
  A)
  \[\left[ \begin{matrix}    6 & -10  \\    4 & 26  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    -10 & 5  \\    4 & 24  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -5 & -6  \\    -4 & -20  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    -5 & -7  \\    -5 & 20  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer15)
  If A and B be two square matrices of order \[\lambda \]whose all the elements are essentially positive integers then the minimum value of \[tr\text{ (}A{{B}^{2}})\] is equal to                    
  A)
  \[{{\lambda }^{3}}\] done clear
  B)
  \[{{\lambda }^{2}}\] done clear
  C)
  \[2{{\lambda }^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer16)
  Matrix A such that \[{{A}^{2}}=2A-I\], where I is the identity matrix, then for \[n\ge 2,\text{ }{{\text{A}}^{n}}\] is equal to
  A)
  \[{{2}^{n-1}}A-(n-1)I\] done clear
  B)
  \[{{2}^{n-1}}A-I\] done clear
  C)
  \[nA-(n-1)I\] done clear
  D)
  \[nA-I\] done clear
  View Solution play_arrow

• question_answer17)
  If \[A=\left[ \begin{matrix}    0 & 1 & 3  \\    1 & 2 & 3  \\    3 & a & 1  \\ \end{matrix} \right]\] and \[{{A}^{-1}}=\left[ \begin{matrix}    1/2 & -1/2 & 1/2  \\    -4 & 3 & c  \\    5/2 & -3/2 & 1/2  \\ \end{matrix} \right]\] Then the value of \[a+c\]is equal to
  A)
   1 done clear
  B)
  0 done clear
  C)
  2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer18)
  Consider the matrices \[A=\left[ \begin{matrix}    4 & 6 & -1  \\    3 & 0 & 2  \\    1 & -2 & 5  \\ \end{matrix} \right],B=\left[ \begin{align}   & \begin{matrix}    2 & 4  \\ \end{matrix} \\  & \begin{matrix}    0 & 1  \\ \end{matrix} \\  & \begin{matrix}    -1 & 2  \\ \end{matrix} \\ \end{align} \right],C=\left[ \begin{matrix}    3  \\    1  \\    2  \\ \end{matrix} \right]\] Out of the given matrix products, which one is not defined.
  A)
  \[{{(AB)}^{T}}C\] done clear
  B)
  \[{{C}^{T}}C{{(AB)}^{T}}\] done clear
  C)
  \[{{C}^{T}}AB\] done clear
  D)
  \[{{A}^{T}}AB{{B}^{T}}C\] done clear
  View Solution play_arrow

• question_answer19)
  If \[A={{[{{a}_{ij}}]}_{n\times n}}\] be a diagonal matrix with diagonal element all different and \[B={{[{{b}_{ij}}]}_{n\times n}}\] be some another matrix. Let \[AB={{[cij]}_{n\times n}}\]then \[{{c}_{ij}}\] is equal to
  A)
  \[{{a}_{jj}}{{b}_{ij}}\] done clear
  B)
  \[{{a}_{ii}}\,{{b}_{ij}}\] done clear
  C)
  \[{{a}_{ij}}\,{{b}_{ij}}\] done clear
  D)
  \[{{a}_{ij}}\,{{b}_{ji}}\] done clear
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• question_answer20)
  If A is any \[2\times 2\] matrix such that \[\left[ \begin{matrix}    1 & 2  \\    0 & 3  \\ \end{matrix} \right]A=\left[ \begin{matrix}    -1 & 0  \\    6 & 3  \\ \end{matrix} \right]\], then what is A equal to?
  A)
  \[\left[ \begin{matrix}    -5 & 1  \\    -2 & 2  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    -5 & -2  \\    1 & 2  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -5 & -2  \\    2 & 1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    5 & 2  \\    -2 & -1  \\ \end{matrix} \right]\] done clear
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• question_answer21)
  Let \[A=\left[ \begin{align}   & \begin{matrix}    5 & 6 & 1  \\ \end{matrix} \\  & \begin{matrix}    2 & -1 & 5  \\ \end{matrix} \\ \end{align} \right]\]. Let there exist a matrix B such that \[AB=\left[ \begin{matrix}    35 & 49  \\    29 & 13  \\ \end{matrix} \right]\]. What is B equal to?
  A)
  \[\left[ \begin{align}   & \begin{matrix}    5 & 1 & 4  \\ \end{matrix} \\  & \begin{matrix}    2 & 6 & 3  \\ \end{matrix} \\ \end{align} \right]\] done clear
  B)
  \[\left[ \begin{align}   & \begin{matrix}    2 & 6 & 3  \\ \end{matrix} \\  & \begin{matrix}    5 & 1 & 4  \\ \end{matrix} \\ \end{align} \right]\] done clear
  C)
  \[\left[ \begin{align}   & \begin{matrix}    5 & 2  \\ \end{matrix} \\  & \begin{matrix}    1 & 6  \\ \end{matrix} \\  & \begin{matrix}    4 & 3  \\ \end{matrix} \\ \end{align} \right]\] done clear
  D)
  \[\left[ \begin{align}   & \begin{matrix}    2 & 5  \\ \end{matrix} \\  & \begin{matrix}    6 & 1  \\ \end{matrix} \\  & \begin{matrix}    3 & 4  \\ \end{matrix} \\ \end{align} \right]\] done clear
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• question_answer22)
  If \[A=\left[ \begin{matrix}    1 & 0  \\    -1 & 7  \\ \end{matrix} \right]\] and \[I=\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\], then the value of k so that \[{{A}^{2}}=8A+kI\] is
  A)
  \[k=7\] done clear
  B)
  \[k=-7\] done clear
  C)
  \[k=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer23)
  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)
  \[\alpha =2ab,\beta ={{a}^{2}}+{{b}^{2}}\] done clear
  B)
  \[\alpha ={{a}^{2}}+{{b}^{2}},\beta =ab\] done clear
  C)
  \[\alpha ={{a}^{2}}+{{b}^{2}},\beta =2ab\] done clear
  D)
  \[\alpha ={{a}^{2}}+{{b}^{2}},\beta ={{a}^{2}}-{{b}^{2}}\] done clear
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• question_answer24)
  Number of square sub-matrices of order 2 (sub-matrix is obtained by deleting appropriate number of rows and columns in a given matrix) that can be formed from the matrix \[\left[ \begin{matrix}    1 & 2 & -1 & 4  \\    2 & 4 & 3 & 5  \\    -1 & -2 & 6 & -7  \\ \end{matrix} \right]\] is
  A)
  12 done clear
  B)
  15    done clear
  C)
  18 done clear
  D)
  \[{{2}^{12}}\] done clear
  View Solution play_arrow

• question_answer25)
  If A is a square matrix such that \[(A-2I)(A+I)=O\]then \[{{A}^{-1}}=\]
  A)
  \[\frac{A-I}{2}\] done clear
  B)
  \[\frac{A+I}{2}\] done clear
  C)
  \[2(A-I)\] done clear
  D)
  \[2A+I\] done clear
  View Solution play_arrow

• question_answer26)
  Let \[A=\left[ \begin{matrix}    1 & 1 & 1  \\    1 & 1 & 1  \\    1 & 1 & 1  \\ \end{matrix} \right]\] be a square matrix of order 3. Then for any positive integer n, what is \[{{A}^{n}}\] equal to?
  A)
  A done clear
  B)
  \[{{3}^{n}}A\] done clear
  C)
  \[({{3}^{n-1}})A\] done clear
  D)
  3A done clear
  View Solution play_arrow

• question_answer27)
  The element \[{{a}_{ij}}\] of square matrix is given by \[{{a}_{ij}}=(i+j)(i-j)\], then matrix A must be
  A)
  Skew-symmetric matrix done clear
  B)
  Triangular matrix done clear
  C)
  Symmetric matrix done clear
  D)
  Null matrix done clear
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• question_answer28)
  If AB = O, then for the matrices \[A=\left[ \begin{matrix}    {{\cos }^{2}}\theta  & \cos \theta \sin \theta   \\    \cos \theta \sin \theta  & {{\sin }^{2}}\theta   \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    {{\cos }^{2}}\phi  & \cos \phi \sin \phi   \\    \cos \phi \sin \phi  & {{\sin }^{2}}\phi   \\ \end{matrix} \right],\theta -\phi \] is
  A)
  An odd number of \[\frac{\pi }{2}\] done clear
  B)
  An odd multiple of \[\pi \] done clear
  C)
  An even multiple of \[\frac{\pi }{2}\] done clear
  D)
              0 done clear
  View Solution play_arrow

• question_answer29)
  If \[A=\left[ \begin{matrix}    2 & 2  \\    2 & 2  \\ \end{matrix} \right]\], then what is \[{{A}^{n}}\] equal to?
  A)
  \[\left[ \begin{matrix}    {{2}^{n}} & {{2}^{n}}  \\    {{2}^{n}} & {{2}^{n}}  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2n & 2n  \\    2n & 2n  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    {{2}^{2n-1}} & {{2}^{2n-1}}  \\    {{2}^{2n-1}} & {{2}^{2n-1}}  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    {{2}^{2n+1}} & {{2}^{2n+1}}  \\    {{2}^{2n+1}} & {{2}^{2n+1}}  \\ \end{matrix} \right]\] done clear
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• question_answer30)
  Let A and B be \[3\times 3\] matrices of real numbers, where A is symmetric, B is skew symmetric, and \[(A+B)(A-B)=(A-B)(A+B).\] If \[{{(AB)}^{t}}={{(-1)}^{k}}AB\]where \[{{(AB)}^{t}}\] is the transpose of the matrix AB, then k is
  A)
  Any integer done clear
  B)
  Odd integer done clear
  C)
  Even integer done clear
  D)
  Cannot say anything done clear
  View Solution play_arrow

• question_answer31)
  If A is a square matrix, then \[A{{A}^{T}}\] is a
  A)
  Skew-symmetric matrix done clear
  B)
  Symmetric matrix done clear
  C)
  Diagonal matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  If \[A=\left[ \begin{matrix}    1 & 2  \\    0 & 3  \\ \end{matrix} \right]\] is a \[2\times 2\] matrix and \[f(x)={{x}^{2}}-x+2\] is a polynomial, then what is f(A)?
  A)
  \[\left[ \begin{matrix}    1 & 7  \\    1 & 7  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2 & 6  \\    0 & 8  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    2 & 6  \\    0 & 6  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    2 & 6  \\    0 & 7  \\ \end{matrix} \right]\] done clear
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• question_answer33)
  If \[P\left[ \begin{matrix}    \cos (\pi /6) & \sin (\pi /6)  \\    -\sin (\pi /6) & \cos (\pi /6)  \\ \end{matrix} \right],A=\left[ \begin{matrix}    1 & 1  \\    0 & 1  \\ \end{matrix} \right]\] and \[Q=PAP'\] then \[P'{{Q}^{2007}}P\] is equal to
  A)
  \[\left[ \begin{matrix}    1 & 2007  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & \sqrt{3}/2  \\    0 & 2007  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    \sqrt{3}/2 & 2007  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    \sqrt{3}/2 & -1/2  \\    1 & 2007  \\ \end{matrix} \right]\] done clear
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• question_answer34)
  If \[A=\left( \begin{matrix}    p & q  \\    0 & 1  \\ \end{matrix} \right)\], then \[{{A}^{8}}=\left( \begin{matrix}    {{p}^{8}} & q\left( \frac{{{p}^{8}}-1}{p-1} \right)  \\    0 & k  \\ \end{matrix} \right)\]. The value of k is
  A)
  1 done clear
  B)
  0    done clear
  C)
  2 done clear
  D)
  -1 done clear
  View Solution play_arrow

• question_answer35)
  If \[A=\left[ \begin{matrix}    1 & 1  \\    1 & 1  \\ \end{matrix} \right]\] then \[{{A}^{100}}\]:
  A)
  \[{{2}^{100}}A\] done clear
  B)
  \[{{2}^{99}}A\] done clear
  C)
  \[{{2}^{101}}A\] done clear
  D)
  None of above done clear
  View Solution play_arrow

• question_answer36)
  If matrix \[A=\left[ \begin{matrix}    -5 & -8 & 0  \\    3 & 5 & 0  \\    1 & 2 & -1  \\ \end{matrix} \right]\] then find\[tr(A)+tr({{A}^{2}})+tr({{A}^{3}})+...+tr({{A}^{100}})\]
  A)
  100 done clear
  B)
  50 done clear
  C)
  200 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer37)
  Let \[A=\left( \begin{matrix}    1 & -1 & 1  \\    2 & 1 & -3  \\    1 & 1 & 1  \\ \end{matrix} \right).\] and 10 \[B=\left( \begin{matrix}    4 & 2 & 2  \\    -5 & 0 & \alpha   \\    1 & -2 & 3  \\ \end{matrix} \right).\] If B is the inverse of matrix A, then \[\alpha \] is
  A)
  5 done clear
  B)
  -1   done clear
  C)
  2 done clear
  D)
  -2 done clear
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• question_answer38)
  If A and B are two matrices such that AB = B and BA = A, then \[{{A}^{2}}+{{B}^{2}}\] is equal to
  A)
  2AB done clear
  B)
  2BA done clear
  C)
  A+B done clear
  D)
  AB done clear
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• question_answer39)
  The values of a, b, c if \[\left[ \begin{matrix}    0 & 2b & c  \\    a & b & -c  \\    a & -b & c  \\ \end{matrix} \right]\] is orthogonal are
  A)
  \[a=\pm \frac{1}{\sqrt{2}};b=\pm \frac{1}{\sqrt{6}};c=\pm \frac{1}{\sqrt{3}}\] done clear
  B)
  \[a=\pm \frac{1}{\sqrt{2}};b=\pm \frac{1}{\sqrt{3}};c=\pm \frac{1}{\sqrt{6}}\] done clear
  C)
  \[a=\pm \frac{1}{\sqrt{6}};b=\pm \frac{1}{\sqrt{2}};c=\pm \frac{1}{\sqrt{3}}\] done clear
  D)
  \[a=\pm \frac{1}{\sqrt{3}};b=\pm \frac{1}{\sqrt{2}};c=\pm \frac{1}{\sqrt{6}}\] done clear
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• question_answer40)
  If \[B=\left[ \begin{matrix}    3 & 4  \\    2 & 3  \\ \end{matrix} \right]\] and \[C=\left[ \begin{matrix}    3 & -4  \\    -2 & 3  \\ \end{matrix} \right]\] and \[X=BC\],find \[{{X}^{n}}\]
  A)
  0 done clear
  B)
  I done clear
  C)
  2I done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer41)
  Let \[A+2B=\left[ \begin{matrix}    1 & 2 & 0  \\    6 & -3 & 3  \\    -5 & 3 & 1  \\ \end{matrix} \right]\] and\[2A-B=\left[ \begin{matrix}    2 & -1 & 5  \\    2 & -1 & 6  \\    0 & 1 & 2  \\ \end{matrix} \right]\], then \[\operatorname{tr}(A)  tr(B)\] is
  A)
  1 done clear
  B)
  3     done clear
  C)
  2 done clear
  D)
  0 done clear
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• question_answer42)
  If \[A=\left[ \begin{matrix}    0 & c & -b  \\    -c & 0 & a  \\    b & -a & 0  \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    {{a}^{2}} & ab & ac  \\    ab & {{b}^{2}} & bc  \\    ac & bc & {{c}^{2}}  \\ \end{matrix} \right]\], then AB is equal to
  A)
  B done clear
  B)
  A    done clear
  C)
  O done clear
  D)
  I done clear
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• question_answer43)
  If the least number of zeroes in a lower triangular matrix is 10, then what is the order of the matrix?
  A)
  \[3\times 3\] done clear
  B)
  \[4\times 4\] done clear
  C)
  \[5\times 5\] done clear
  D)
  \[10\times 10\] done clear
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• question_answer44)
  If a matrix A is such that\[3{{A}^{3}}+2{{A}^{2}}+5A+I=0,\] then what is \[{{A}^{-1}}\] equal to?
  A)
  \[-(3{{A}^{2}}+2A+5I)\] done clear
  B)
  \[3{{A}^{2}}+2A+5I\] done clear
  C)
  \[3{{A}^{2}}-2A-5I\] done clear
  D)
  \[(3{{A}^{2}}+2A-5I)\] done clear
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• question_answer45)
  If A and B are symmetric matrices of the same order and X = AB + BA and Y = AB - BA, then \[{{(XY)}^{T}}\] is equal to
  A)
  XY done clear
  B)
  YX done clear
  C)
  -YX done clear
  D)
  None of these done clear
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• question_answer46)
  If \[A=\left[ \begin{matrix}    0 & 1  \\    0 & 0  \\ \end{matrix} \right]\], I is the unit matrix of order 2 and a, b are arbitrary constants, then \[{{(aI+bA)}^{2}}\] is equal to
  A)
  \[{{a}^{2}}I+abA\] done clear
  B)
  \[{{a}^{2}}I+2abA\] done clear
  C)
  \[{{a}^{2}}I+{{b}^{2}}A\] done clear
  D)
  None of these done clear
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• question_answer47)
  If \[{{B}^{n}}-A=I\]and \[A=\left[ \begin{matrix}    26 & 26 & 18  \\    25 & 37 & 17  \\    52 & 39 & 50  \\ \end{matrix} \right],B=\left[ \begin{matrix}    1 & 4 & 2  \\    3 & 5 & 1  \\    7 & 1 & 6  \\ \end{matrix} \right]\] then n =
  A)
  2 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  5 done clear
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• question_answer48)
  If A is symmetric as well as skew-symmetric matrix, then A is
  A)
  Diagonal done clear
  B)
  Null done clear
  C)
  Triangular done clear
  D)
  None of these done clear
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• question_answer49)
  If \[A=\left[ \begin{matrix}    \cos \theta  & \sin \theta   \\    -\sin \theta  & \cos \theta   \\ \end{matrix} \right]\] then \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}{{A}^{n}}\] is
  A)
  A null matrix done clear
  B)
  An identity matrix done clear
  C)
  \[\left[ \begin{matrix}    0 & 1  \\    -1 & 0  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
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• question_answer50)
  Let A, B, C, D be (not necessarily square) real matrices such that \[{{A}^{T}}=BCD;\text{ }{{B}^{T}}=CDA;\] \[{{C}^{T}}=DAB\] and \[{{D}^{T}}=ABC\] for the matrix \[S=ABCD,{{S}^{3}}=\]
  A)
  I done clear
  B)
  \[{{S}^{2}}\]   done clear
  C)
  S done clear
  D)
  O done clear
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• question_answer51)
  If \[A=\left[ \begin{matrix}    \alpha  & \beta   \\    \gamma  & \delta   \\ \end{matrix} \right]\] such that \[{{A}^{2}}\] is a two - rowed unit matrix, then \[\delta \] is equal to
  A)
  \[\alpha \] done clear
  B)
  \[\beta \] done clear
  C)
  \[\gamma \] done clear
  D)
  None of these done clear
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• question_answer52)
  Which of the following is/are correct?
  A)
  B' AB is symmetric if A is symmetric done clear
  B)
  B' AB is skew-symmetric if A is symmetric done clear
  C)
  B' AB is symmetric if A is skew-symmetric done clear
  D)
  None of these done clear
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• question_answer53)
  If Z is an idempotent matrix, then \[{{(I+Z)}^{n}}\]
  A)
  \[I+{{2}^{n}}Z\] done clear
  B)
  \[I+({{2}^{n}}-1)Z\] done clear
  C)
  \[I-({{2}^{n}}-1)Z\] done clear
  D)
  None of these done clear
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• question_answer54)
  If A and B are square matrices of size \[n\times n\] such that \[{{A}^{2}}-{{B}^{2}}=(A-B)(A+B)\], then which of the following will be always true?
  A)
  A = B done clear
  B)
  AB = BA done clear
  C)
  Either of A or B is a zero matrix done clear
  D)
  Either of A or B is identity matrix done clear
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• question_answer55)
  Consider the following in respect of the matrix \[A=\left( \begin{matrix}    -1 & 1  \\    1 & -1  \\ \end{matrix} \right):\] 1. \[{{A}^{2}}=-A\] 2. \[{{A}^{3}}=4A\] Which of the above is/are correct?
  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2 done clear
  D)
  Neither 1 nor 2 done clear
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• question_answer56)
  If \[A=\left[ \begin{matrix}    \alpha  & 0  \\    1 & 1  \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    9 & a  \\    b & c  \\ \end{matrix} \right]\] and \[{{A}^{2}}=B\], then the value of a + b + c is
  A)
  1 or -1 done clear
  B)
  5 or -1 done clear
  C)
  5 or 1 done clear
  D)
  no real values done clear
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• question_answer57)
  If \[X=\left[ \begin{matrix}    1 & -2  \\    0 & 3  \\ \end{matrix} \right]\], and I is a \[2\times 2\] identity matrix, then \[{{X}^{2}}-2X+3I\] equals to which one of the following?
  A)
  -I done clear
  B)
  -2X done clear
  C)
  2X done clear
  D)
  4X done clear
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• question_answer58)
  Let \[A=\left[ \begin{matrix}    0 & \alpha   \\    0 & 0  \\ \end{matrix} \right]\] and \[{{(A+I)}^{50}}-50A=\left[ \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right],\]find\[abc+abd+bcd+acd\]   
  A)
  0 done clear
  B)
  -1 done clear
  C)
  1 done clear
  D)
  None of these done clear
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• question_answer59)
  If \[\left[ \begin{matrix}    2 & 0 & 7  \\    0 & 1 & 0  \\    1 & -2 & 1  \\ \end{matrix} \right]\left[ \begin{matrix}    -x & 14x & 7x  \\    0 & 1 & 0  \\    x & -4x & -2x  \\ \end{matrix} \right]=\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 0  \\    0 & 0 & 1  \\ \end{matrix} \right]\]then find the value of x
  A)
  \[\frac{1}{2}\] done clear
  B)
              \[\frac{1}{5}\] done clear
  C)
  No unique value of 'x' done clear
  D)
  None of these done clear
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• question_answer60)
  \[A=\left[ \begin{matrix}    1 & -1  \\    2 & 3  \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    2 & 3  \\    -1 & -2  \\ \end{matrix} \right]\] , then which of  the following is/are correct? 1. \[AB({{A}^{-1}}{{B}^{-1}})\] is a unit matrix. 2. \[{{(AB)}^{-1}}={{A}^{-1}}{{B}^{-1}}\] Select the correct answer using the code given below:
  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 only 2 done clear
  D)
  Neither 1 nor 2 done clear
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