JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

done Types of matrices, Algebra of matrices Total Questions - 107

• question_answer1)
  In a skew symmetric matrix, the diagonal elements are all [MP PET 1987]
  A)
  Different from each other done clear
  B)
  Zero done clear
  C)
  One done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  If \[M=\left[ \begin{matrix}    1 & 2  \\    2 & 3  \\ \end{matrix} \right]\]and \[{{M}^{2}}-\lambda M-{{I}_{2}}=0\], then \[\lambda =\] [MP PET 1990, 2001]
  A)
  - 2 done clear
  B)
  2 done clear
  C)
  - 4 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer3)
  If \[A=\left[ \begin{matrix}    \cos \alpha  & -\sin \alpha   \\    \sin \alpha  & \cos \alpha   \\ \end{matrix} \right]\]and \[B=\left[ \begin{matrix}    \cos \beta  & -\sin \beta   \\    \sin \beta  & \cos \beta   \\ \end{matrix} \right]\], then the correct relation is
  A)
  \[{{A}^{2}}={{B}^{2}}\] done clear
  B)
  \[A+B=B-A\] done clear
  C)
  \[AB=BA\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer4)
  If \[A=\left[ \begin{matrix}    1 & 0 & 1  \\    0 & 1 & 1  \\    1 & 0 & 0  \\ \end{matrix} \right]\], then A is [MP PET 1991]
  A)
  Symmetric done clear
  B)
  Skew-symmetric done clear
  C)
  Non-singular done clear
  D)
  Singular done clear
  View Solution play_arrow

• question_answer5)
  If \[A=\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 0  \\    a & b & -1  \\ \end{matrix} \right]\], then \[{{A}^{2}}=\] [MNR 1980; Pb. CET 1990; DCE 2001]
  A)
  Unit matrix done clear
  B)
  Null matrix done clear
  C)
  A done clear
  D)
  - A done clear
  View Solution play_arrow

• question_answer6)
  If \[A=\left[ \begin{matrix}    1 & 1  \\    0 & 1  \\ \end{matrix} \right]\], then \[{{A}^{n}}=\] [RPET 1995]
  A)
  \[\left[ \begin{matrix}    1 & n  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    n & n  \\    0 & n  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    n & 1  \\    0 & n  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 1  \\    0 & n  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer7)
  \[AB=0\], if and only if   [MNR 1981; Karnataka CET 1993]
  A)
  \[A\ne O,B=O\] done clear
  B)
  \[A=O,B\ne O\] done clear
  C)
  \[A=O\]or \[B=O\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer8)
  If \[A=\left[ \begin{matrix}    2 & 2  \\    a & b  \\ \end{matrix} \right]\]and  \[{{A}^{2}}=O\], then \[(a,b)=\]
  A)
  \[(-2,\,-2)\] done clear
  B)
  \[(2,\,-2)\] done clear
  C)
  \[(-2,\,2)\] done clear
  D)
  \[(2,\,2)\] done clear
  View Solution play_arrow

• question_answer9)
  If [m  n]\[\left[ \begin{matrix}    m  \\    n  \\ \end{matrix} \right]=[25]\] and m< n,  then (m, n) =
  A)
  (2, 3) done clear
  B)
  (3, 4) done clear
  C)
  (4, 3) done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer10)
  \[A=\left[ \begin{matrix}    4 & 6 & -1  \\    3 & 0 & 2  \\    1 & -2 & 5  \\ \end{matrix} \right]\],\[B=\left[ \begin{matrix}    2 & 4  \\    0 & 1  \\    -1 & 2  \\ \end{matrix} \right],\,\,C=\left[ \begin{matrix}    3  \\    1  \\    2  \\ \end{matrix} \right]\], then the expression which is not defined is [MP PET 1987]
  A)
  \[{{A}^{2}}+2B-2A\] done clear
  B)
  \[C{C}'\] done clear
  C)
  \[{B}'C\] done clear
  D)
  \[AB\] done clear
  View Solution play_arrow

• question_answer11)
  If \[A=\left[ \begin{matrix}    i & 0  \\    0 & -i  \\ \end{matrix} \right],B=\left[ \begin{matrix}    0 & i  \\    i & 0  \\ \end{matrix} \right]\], where \[i=\sqrt{-1}\], then the correct relation is
  A)
  \[A+B=O\] done clear
  B)
  \[{{A}^{2}}={{B}^{2}}\] done clear
  C)
  \[A-B=O\] done clear
  D)
  \[{{A}^{2}}+{{B}^{2}}=O\] done clear
  View Solution play_arrow

• question_answer12)
  If the matrix \[\left[ \begin{matrix}    1 & 3 & \lambda +2  \\    2 & 4 & 8  \\    3 & 5 & 10  \\ \end{matrix} \right]\]is singular, then \[\lambda =\]  [MP PET 1990; Pb. CET 2000]
  A)
  - 2 done clear
  B)
  4 done clear
  C)
  2 done clear
  D)
  - 4 done clear
  View Solution play_arrow

• question_answer13)
  If \[A=\left[ \begin{matrix}    ab & {{b}^{2}}  \\    -{{a}^{2}} & -ab  \\ \end{matrix} \right]\]and \[{{A}^{n}}=O\], then the minimum value of n is
  A)
  2 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer14)
  If \[A=\left[ \begin{matrix}    1/3 & 2  \\    0 & 2x-3  \\ \end{matrix} \right],B=\left[ \begin{matrix}    3 & 6  \\    0 & -1  \\ \end{matrix} \right]\]and \[AB=I\], then x = [MP PET 1987]
  A)
  -1 done clear
  B)
  1 done clear
  C)
  0 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer15)
  If \[AB=C\], then matrices \[A,B,C\]are [MP PET 1991]
  A)
  \[{{A}_{2\times 3}},{{B}_{3\times 2}},{{C}_{2\times 3}}\] done clear
  B)
  \[{{A}_{3\times 2}},{{B}_{2\times 3}},{{C}_{3\times 2}}\] done clear
  C)
  \[{{A}_{3\times 3}},{{B}_{2\times 3}},{{C}_{3\times 3}}\] done clear
  D)
  \[{{A}_{3\times 2}},{{B}_{2\times 3}},{{C}_{3\times 3}}\] done clear
  View Solution play_arrow

• question_answer16)
  If \[A=\left[ \begin{matrix}    \lambda  & 1  \\    -1 & -\lambda   \\ \end{matrix} \right]\], then for what value of \[\lambda ,\,{{A}^{2}}=O\]  [MP PET 1992]
  A)
  0 done clear
  B)
  \[\pm \text{ }1\] done clear
  C)
  - 1 done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer17)
  If \[A=\left[ \begin{matrix}    0 & 1 & -2  \\    -1 & 0 & 5  \\    2 & -5 & 0  \\ \end{matrix} \right]\], then [MNR 1982]
  A)
  \[{A}'=A\] done clear
  B)
  \[{A}'=-A\] done clear
  C)
  \[{A}'=2A\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer18)
  If \[A=\left[ \begin{matrix}    4 & 1  \\    3 & 2  \\ \end{matrix} \right]\]and \[I=\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\], \[{{A}^{2}}-6A=\] [MP PET 1987]
  A)
  3I done clear
  B)
  5I done clear
  C)
  - 5I done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  If \[A=\left[ \begin{matrix}    1 & 2 & 3  \\    5 & 0 & 7  \\    6 & 2 & 5  \\ \end{matrix} \right]\]and \[B=\left[ \begin{matrix}    1 & 3 & 5  \\    0 & 0 & 2  \\ \end{matrix} \right]\], then which of the following is defined
  A)
  AB done clear
  B)
  \[A+B\] done clear
  C)
  \[{A}'{B}'\] done clear
  D)
  \[{B}'{A}'\] done clear
  View Solution play_arrow

• question_answer20)
  If \[A=[1\,\,2\,\text{ }3]\]and \[B=\left[ \begin{matrix}    -5 & 4 & 0  \\    0 & 2 & -1  \\    1 & -3 & 2  \\ \end{matrix} \right]\], then \[AB=\] [MP PET 1988]
  A)
  \[\left[ \begin{matrix}    -5 & 4 & 0  \\    0 & 4 & -2  \\    3 & -9 & 6  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{align}   & 3 \\  & 1 \\  & 1 \\ \end{align} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -2 & -1 & 4  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    -5 & 8 & 0  \\    0 & 4 & -3  \\    1 & -6 & 6  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer21)
  If A is a \[m\times n\]matrix and B is a matrix such that both AB and BA  are defined, then the order of B is
  A)
  \[m\times n\] done clear
  B)
  \[n\times m\] done clear
  C)
  \[m\times m\] done clear
  D)
  \[n\times n\] done clear
  View Solution play_arrow

• question_answer22)
  If \[A=\left[ \begin{align}   & 1 \\  & 2 \\  & 3 \\ \end{align} \right],\]then \[A{A}'=\] [MP PET 1992]
  A)
  14 done clear
  B)
    \[\left[ \begin{align}   & 1 \\  & 4 \\  & 3 \\ \end{align} \right]\] done clear
  C)
   \[\left[ \begin{matrix}    1 & 2 & 3  \\    2 & 4 & 6  \\    3 & 6 & 9  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer23)
  If \[\left[ \begin{matrix}    2 & -3  \\    4 & 0  \\ \end{matrix} \right]-\left[ \begin{matrix}    a & c  \\    b & d  \\ \end{matrix} \right]=\left[ \begin{matrix}    1 & 4  \\    2 & -5  \\ \end{matrix} \right]\], then \[(a,b,c,d)=\]
  A)
  \[(1,\,6,\,2,\,5)\] done clear
  B)
  (1, 2, 7, 5) done clear
  C)
  (1, 2, -7, 5) done clear
  D)
  (-1, -2, 7, -5) done clear
  View Solution play_arrow

• question_answer24)
  If \[A=\left[ \begin{matrix}    -1 & 0 & 0  \\    0 & -1 & 0  \\    0 & 0 & -1  \\ \end{matrix} \right]\], then \[{{A}^{2}}\]is  [MNR 1980]
  A)
  Null matrix done clear
  B)
   Unit matrix done clear
  C)
    A done clear
  D)
  2A done clear
  View Solution play_arrow

• question_answer25)
  If \[A=\left[ \begin{matrix}    0 & 2 & 0  \\    0 & 0 & 3  \\    -2 & 2 & 0  \\ \end{matrix} \right]\]and \[B=\left[ \begin{matrix}    1 & 2 & 3  \\    3 & 4 & 5  \\    5 & -4 & 0  \\ \end{matrix} \right]\], then the element of 3rd row and third column in AB will be
  A)
  - 18 done clear
  B)
    4 done clear
  C)
    - 12 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer26)
  If \[A=\left[ \begin{matrix}    2 & 0 & 0  \\    0 & 2 & 0  \\    0 & 0 & 2  \\ \end{matrix} \right],\]then \[{{A}^{5}}=\] [MP PET 1995,1999; Pb. CET 2000]
  A)
  5A done clear
  B)
  10A done clear
  C)
  16A done clear
  D)
  32A done clear
  View Solution play_arrow

• question_answer27)
  If \[A=\left[ \begin{matrix}    0 & 1  \\    0 & 0  \\ \end{matrix} \right]\]and \[AB=O\], then B = [MP PET 1989]
  A)
  \[\left[ \begin{matrix}    1 & 1  \\    1 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    0 & 1  \\    -1 & 0  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    0 & -1  \\    1 & 0  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    -1 & 0  \\    0 & 0  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer28)
  If A and B are square matrices of order 2, then \[{{(A+B)}^{2}}=\] [MP PET 1992]
  A)
  \[{{A}^{2}}+2AB+{{B}^{2}}\] done clear
  B)
  \[{{A}^{2}}+2AB+{{B}^{2}}\] done clear
  C)
  \[{{A}^{2}}+2BA+{{B}^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer29)
  If \[A=\left[ \begin{matrix}    1 & 0 & 0 & 0  \\    2 & 3 & 0 & 0  \\    4 & 5 & 6 & 0  \\    7 & 8 & 9 & 10  \\ \end{matrix} \right]\], then A is
  A)
  An upper triangular matrix done clear
  B)
  A null matrix done clear
  C)
  A lower triangular matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer30)
  Square matrix \[{{[{{a}_{ij}}]}_{n\times n}}\]will be an upper triangular matrix, if
  A)
  \[{{a}_{ij}}\ne \]0, for \[i>j\] done clear
  B)
  \[{{a}_{ij}}\ne \], for \[i>j\] done clear
  C)
  \[{{a}_{ij}}=\]0, for \[i<j\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer31)
  If the matrix \[\left[ \begin{matrix}    0 & 1 & -2  \\    -1 & 0 & 3  \\    \lambda  & -3 & 0  \\ \end{matrix} \right]\]is singular, then \[\lambda \]= [MP PET 1989]
  A)
  - 2 done clear
  B)
  - 1 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer32)
  Which of the following relations is incorrect
  A)
  \[(A+B+....+l{)}'={A}'+{B}'+....+{l}'\] done clear
  B)
  \[(AB....l{)}'={A}'{B}'....{l}'\] done clear
  C)
  \[(kA{)}'=k{A}'\] done clear
  D)
  \[(A{)}'=A\] done clear
  View Solution play_arrow

• question_answer33)
  If A is a square matrix of order n and \[A=k\]B, where k is a scalar, then |A|= [Karnataka CET 1992]
  A)
  |B| done clear
  B)
  \[k|B|\] done clear
  C)
  \[{{k}^{n}}\]|B| done clear
  D)
  \[n|B|\] done clear
  View Solution play_arrow

• question_answer34)
  If \[A=dig(2,-1,\,3),B=dig(-1,\,3,\,2)\], then \[{{A}^{2}}B=\]
  A)
  dig (5, 4, 11) done clear
  B)
  dig (-4, 3, 18) done clear
  C)
  dig (3, 1, 8 ) done clear
  D)
  B done clear
  View Solution play_arrow

• question_answer35)
   \[\cos \theta \left[ \begin{matrix}    \cos \theta  & \sin \theta   \\    -\sin \theta  & \cos \theta   \\ \end{matrix} \right]+\sin \theta \left[ \begin{matrix}    \sin \theta  & -\cos \theta   \\    \cos \theta  & \sin \theta   \\ \end{matrix} \right]=\]
  A)
  \[\left[ \begin{matrix}    0 & 0  \\    0 & 0  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & 0  \\    0 & 0  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    0 & 1  \\    1 & 0  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer36)
  If I  is a unit matrix, then 3I will be
  A)
  A unit matrix done clear
  B)
  A triangular matrix done clear
  C)
  A scalar matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer37)
  If \[A=[a\,\,b],B=[-b-a]\]and \[C=\left[ \begin{align}   & \,\,\,\,a \\  & -a \\ \end{align} \right]\], then the correct statement is [AMU 1987]
  A)
  \[A=-B\] done clear
  B)
  \[A+B=A-B\] done clear
  C)
  \[AC=BC\] done clear
  D)
  \[CA=CB\] done clear
  View Solution play_arrow

• question_answer38)
  If \[A=\left[ \begin{matrix}    1 & a  \\    0 & 1  \\ \end{matrix} \right]\], then \[{{A}^{4}}\]is equal to [MP PET 1993; Pb. CET 2001]
  A)
  \[\left[ \begin{matrix}    1 & {{a}^{4}}  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    4 & 4a  \\    0 & 4  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    4 & {{a}^{4}}  \\    0 & 4  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 4a  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer39)
  If \[\left[ \begin{matrix}    3 & 1  \\    4 & 1  \\ \end{matrix} \right]X=\left[ \begin{matrix}    5 & -1  \\    2 & 3  \\ \end{matrix} \right],\]then X = [MP PET 1994]
  A)
  \[\left[ \begin{matrix}    -3 & 4  \\    14 & -13  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    3 & -4  \\    -14 & 13  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    3 & 4  \\    14 & 13  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    -3 & 4  \\    -14 & 13  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer40)
  Which of the following is incorrect
  A)
  \[{{A}^{2}}-{{B}^{2}}=(A+B)(A-B)\] done clear
  B)
  \[{{({{A}^{T}})}^{T}}=A\] done clear
  C)
  \[{{(AB)}^{n}}={{A}^{n}}{{B}^{n}},\]where A, B commute done clear
  D)
  \[(A-I)(I+A)=O\Leftrightarrow {{A}^{2}}=I\] done clear
  View Solution play_arrow

• question_answer41)
  \[A,B\] are n-rowed square matrices such that \[AB=O\]and B is non-singular. Then
  A)
  \[A\ne O\] done clear
  B)
  \[A=O\] done clear
  C)
  \[A=I\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer42)
  The order of \[[x\,y\,z]\,\,\left[ \begin{matrix}    a & h & g  \\    h & b & f  \\    g & f & c  \\ \end{matrix} \right]\,\left[ \begin{align}   & x \\  & y \\  & z \\ \end{align} \right]\] is  [EAMCET 1994]
  A)
  \[3\times 1\] done clear
  B)
  \[1\times 1\] done clear
  C)
  \[1\times 3\] done clear
  D)
  \[3\times 3\] done clear
  View Solution play_arrow

• question_answer43)
  If A and B are two matrices such that \[AB=B\]and \[BA=A,\] then \[{{A}^{2}}+{{B}^{2}}=\] [EAMCET 1994]
  A)
  \[2AB\] done clear
  B)
  \[2BA\] done clear
  C)
  \[A+B\] done clear
  D)
  \[AB\] done clear
  View Solution play_arrow

• question_answer44)
  If A and B are two matrices and \[(A+B)(A-B)\]\[={{A}^{2}}-{{B}^{2}}\], then [RPET 1995]
  A)
  \[AB=BA\] done clear
  B)
  \[{{A}^{2}}+{{B}^{2}}={{A}^{2}}-{{B}^{2}}\] done clear
  C)
  \[{A}'{B}'=AB\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer45)
  \[A=\left[ \begin{matrix}    5 & -3  \\    2 & 4  \\ \end{matrix} \right]\]and \[B=\left[ \begin{matrix}    6 & -4  \\    3 & 6  \\ \end{matrix} \right],\]then \[A-B=\]  [RPET 1995]
  A)
  \[\left[ \begin{matrix}    11 & -7  \\    5 & 10  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    -1 & \text{   }1  \\    -1 & -2  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    11 & 7  \\    5 & -10  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    12 & -7  \\    5 & -10  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer46)
  If \[X=\left[ \begin{matrix}    3 & -4  \\    1 & -1  \\ \end{matrix} \right]\], then the value of \[{{X}^{n}}\]is [EAMCET 1991]
  A)
  \[\left[ \begin{matrix}    3n & -4n  \\    n & -n  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2+n & 5-n  \\    n & -n  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    {{3}^{n}} & {{(-4)}^{n}}  \\    {{1}^{n}} & {{(-1)}^{n}}  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer47)
  If \[A=\left[ \begin{matrix}    i & 0  \\    0 & i  \\ \end{matrix} \right]\], then \[{{A}^{2}}=\] [EAMCET 1983]
  A)
  \[\left[ \begin{matrix}    1 & 0  \\    0 & -1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    -1 & 0  \\    0 & -1  \\ \end{matrix} \right]\] 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_answer48)
  If \[A\] and \[B\] are square matrices of same order, then [Roorkee 1995]
  A)
  \[A+B=B+A\] done clear
  B)
  \[A+B=A-B\] done clear
  C)
  \[A-B=B-A\] done clear
  D)
  \[AB=BA\] done clear
  View Solution play_arrow

• question_answer49)
  If \[A=\left[ \begin{matrix}    0 & 1  \\    1 & 0  \\ \end{matrix} \right],\]then \[{{A}^{4}}\]= [EAMCET 1994]
  A)
  \[\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & 1  \\    0 & 0  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    0 & 0  \\    1 & 1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    0 & 1  \\    1 & 0  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer50)
  If \[A=\left[ \begin{matrix}    3 & 1  \\    -1 & 2  \\ \end{matrix} \right]\], then \[{{A}^{2}}=\] [Karnataka CET 1994]
  A)
  \[\left[ \begin{matrix}    8 & -5  \\    -5 & 3  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    8 & -5  \\    5 & 3  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    8 & -5  \\    -5 & -3  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    8 & 5  \\    -5 & 3  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer51)
  If A and B are two matrices such that A+B and AB are both defined, then [Pb. CET 1990]
  A)
  \[A\]and B are two matrices not necessarily of same order done clear
  B)
  A and B are square matrices of same order done clear
  C)
  Number of columns of A= Number of rows of B done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer52)
  If \[A=\left[ \begin{matrix}    1 & 3 & 0  \\    -1 & 2 & 1  \\    0 & 0 & 2  \\ \end{matrix} \right],B=\left[ \begin{matrix}    2 & 3 & 4  \\    1 & 2 & 3  \\    -1 & 1 & 2  \\ \end{matrix} \right]\], then AB = [EAMCET 1987]
  A)
  \[\left[ \begin{matrix}    5 & 9 & 13  \\    -1 & 2 & 4  \\    -1 & 2 & 4  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    5 & 9 & 13  \\    -1 & 2 & 4  \\    -2 & 2 & 4  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 2 & 4  \\    -1 & 2 & 4  \\    -2 & 2 & 4  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer53)
  If  \[A=\left[ \begin{matrix}    1 & 1  \\    0 & 1  \\ \end{matrix} \right],\]\[B=\left[ \begin{matrix}    0 & 1  \\    1 & 0  \\ \end{matrix} \right],\]then \[AB=\] [EAMCET 1984]
  A)
  \[\left[ \begin{matrix}    0 & 0  \\    0 & 0  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & 1  \\    1 & 0  \\ \end{matrix} \right]\] 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_answer54)
  If \[A=\left[ \begin{matrix}    i & 0  \\    0 & i  \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    0 & -i  \\    -i & 0  \\ \end{matrix} \right]\], then \[(A+B)(A-B)\] is equal to [RPET 1994]
  A)
  \[{{A}^{2}}-{{B}^{2}}\] done clear
  B)
  \[{{A}^{2}}+{{B}^{2}}\] done clear
  C)
  \[{{A}^{2}}-{{B}^{2}}+BA+AB\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer55)
  If \[A=\left[ \begin{matrix}    1 & -2  \\    3 & 0  \\ \end{matrix} \right],\] \[B=\left[ \begin{matrix}    -1 & 4  \\    2 & 3  \\ \end{matrix} \right]\], \[C=\left[ \begin{matrix}    0 & -1  \\    1 & 0  \\ \end{matrix} \right]\], then \[5A-3B-2C\]= [RPET 1992, 94]
  A)
  \[\left[ \begin{matrix}    8 & 20  \\    7 & 9  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    8 & -20  \\    7 & -9  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -8 & 20  \\    -7 & 9  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    8 & 7  \\    -20 & -9  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer56)
  If \[\left[ \begin{matrix}    x & 0  \\    1 & y  \\ \end{matrix} \right]+\left[ \begin{matrix}    -2 & 1  \\    3 & 4  \\ \end{matrix} \right]=\left[ \begin{matrix}    3 & 5  \\    6 & 3  \\ \end{matrix} \right]-\left[ \begin{matrix}    2 & 4  \\    2 & 1  \\ \end{matrix} \right]\], then [RPET 1994]
  A)
  \[x=-3,y=-2\] done clear
  B)
  \[x=3,y=-2\] done clear
  C)
  \[x=3,y=2\] done clear
  D)
  \[x=-3,y=2\] done clear
  View Solution play_arrow

• question_answer57)
  If \[A=\left[ \begin{matrix}    x & 1  \\    1 & 0  \\ \end{matrix} \right]\]and \[{{A}^{2}}\] is the identity matrix, then x = [EAMCET 1993]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer58)
  If \[A=\left[ \begin{matrix}    0 & 1  \\    1 & 0  \\ \end{matrix} \right],B=\left[ \begin{matrix}    0 & -i  \\    i & 0  \\ \end{matrix} \right]\] then \[{{(A+B)}^{2}}\]equals [RPET 1994]
  A)
  \[{{A}^{2}}+{{B}^{2}}\] done clear
  B)
  \[{{A}^{2}}+{{B}^{2}}+2AB\] done clear
  C)
  \[{{A}^{2}}+{{B}^{2}}+AB-BA\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer59)
  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 [RPET 1992]
  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
  View Solution play_arrow

• question_answer60)
  Matrix theory was introduced by
  A)
  Newton done clear
  B)
  Cayley-Hamilton done clear
  C)
  Cauchy done clear
  D)
  Euclid done clear
  View Solution play_arrow

• question_answer61)
  If \[A=\left[ \begin{matrix}    1 & 2  \\    -3 & 0  \\ \end{matrix} \right]\]and \[B=\left[ \begin{matrix}    -1 & 0  \\    2 & 3  \\ \end{matrix} \right]\], then  [MP PET 1996]
  A)
  \[{{A}^{2}}=A\] done clear
  B)
  \[{{B}^{2}}=B\] done clear
  C)
  \[AB\ne BA\] done clear
  D)
  \[AB=BA\] done clear
  View Solution play_arrow

• question_answer62)
  Which one of the following is not true [Kurukshetra CEE 1998]
  A)
  Matrix addition is commutative done clear
  B)
  Matrix addition is associative done clear
  C)
  Matrix multiplication is commutative done clear
  D)
  Matrix multiplication is associative done clear
  View Solution play_arrow

• question_answer63)
  In order that the matrix \[\left[ \begin{matrix}    1 & 2 & 3  \\    4 & 5 & 6  \\    3 & \lambda  & 5  \\ \end{matrix} \right]\] be non-singular, \[\lambda \] should not be equal to  [Kurukshetra CEE 1998]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer64)
  If \[U=[2\,-3\,\,4],X=[0\,\,2\,\,3],\] \[V=\left[ \begin{align}   & 3 \\  & 2 \\  & 1 \\ \end{align} \right]\] and \[Y=\left[ \begin{align}   & 2 \\  & 2 \\  & 4 \\ \end{align} \right]\], then \[UV+XY\]= [MP PET 1997]
  A)
  20 done clear
  B)
  [- 20] done clear
  C)
  - 20 done clear
  D)
  [20] done clear
  View Solution play_arrow

• question_answer65)
  If \[A=\left[ \begin{matrix}    0 & i  \\    -i & 0  \\ \end{matrix} \right]\], then the value of \[{{A}^{40}}\]is  [RPET 1999]
  A)
  \[\left[ \begin{matrix}    0 & 1  \\    1 & 0  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 1  \\    0 & 0  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    -1 & 1  \\    0 & -1  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer66)
  If \[A=\left[ \begin{matrix}    1 & -2 & 1  \\    2 & 1 & 3  \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    2 & 1  \\    3 & 2  \\    1 & 1  \\ \end{matrix} \right]\], then \[{{(AB)}^{T}}=\] [RPET 1996]
  A)
  \[\left[ \begin{matrix}    -3 & -2  \\    10 & 7  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    -3 & 10  \\    -2 & 7  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -3 & 10  \\    7 & -2  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    3 & 10  \\    2 & 7  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer67)
  If \[A=[1\,2\,3],B=\left[ \begin{align}   & 2 \\  & 3 \\  & 4 \\ \end{align} \right]\] and \[C=\left[ \begin{matrix}    1 & 5  \\    0 & 2  \\ \end{matrix} \right]\], then which of the following is defined [RPET 1996]
  A)
  AB done clear
  B)
  \[BA\] done clear
  C)
  \[(AB)\,\text{. }C\] done clear
  D)
  \[(AC)\,.\,B\] done clear
  View Solution play_arrow

• question_answer68)
  The matrix product \[AB=O\], then             [Kurukshetra CEE 1998; RPET 2001]
  A)
  \[A=O\]and \[B=O\] done clear
  B)
  \[A=O\]or \[B=O\] done clear
  C)
  A is null matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer69)
  If A and B are square matrices of order n × n, then \[{{(A-B)}^{2}}\] is equal to        [Karnataka CET 1999; Kerala (Engg.) 2002]
  A)
  \[{{A}^{2}}-{{B}^{2}}\] done clear
  B)
  \[{{A}^{2}}-2AB+{{B}^{2}}\] done clear
  C)
  \[{{A}^{2}}+2AB+{{B}^{2}}\] done clear
  D)
  \[{{A}^{2}}-AB-BA+{{B}^{2}}\] done clear
  View Solution play_arrow

• question_answer70)
  Choose the correct answer [Karnataka CET 1999]
  A)
  Every identity matrix is a scalar matrix done clear
  B)
  Every scalar matrix is an identity matrix done clear
  C)
  Every diagonal matrix is an identity matrix done clear
  D)
  A square matrix whose each element is 1 is an identity matrix done clear
  View Solution play_arrow

• question_answer71)
  If \[A=\left[ \begin{matrix}    1 & 0  \\    2 & 0  \\ \end{matrix} \right],B=\left[ \begin{matrix}    0 & 0  \\    1 & 12  \\ \end{matrix} \right]\], then [DCE 1999]
  A)
  \[AB=O,BA=O\] done clear
  B)
  \[AB=O,BA\ne O\] done clear
  C)
  \[AB\ne O,BA=O\] done clear
  D)
  \[AB\ne O,BA\ne O\] done clear
  View Solution play_arrow

• question_answer72)
  The matrix \[\left[ \begin{matrix}    2 & 5 & -7  \\    0 & 3 & 11  \\    0 & 0 & 9  \\ \end{matrix} \right]\]is known as [Karnataka CET 1999; Pb. CET 2001]
  A)
  Symmetric matrix done clear
  B)
  Diagonal matrix done clear
  C)
  Upper triangular matrix done clear
  D)
  Skew symmetric matrix done clear
  View Solution play_arrow

• question_answer73)
  If \[A=\left( \begin{matrix}    i & 1  \\    0 & i  \\ \end{matrix} \right)\], then \[{{A}^{4}}\]equals [AMU 1999]
  A)
  \[\left( \begin{matrix}    1 & -4i  \\    0 & 1  \\ \end{matrix} \right)\] done clear
  B)
  \[\left( \begin{matrix}    -1 & -4i  \\    0 & -1  \\ \end{matrix} \right)\] done clear
  C)
  \[\left( \begin{matrix}    -i & 4  \\    0 & i  \\ \end{matrix} \right)\] done clear
  D)
  \[\left( \begin{matrix}    1 & 4  \\    0 & 1  \\ \end{matrix} \right)\] done clear
  View Solution play_arrow

• question_answer74)
  \[\left[ \begin{align}   & \,\,\,1 \\  & -1 \\  & \,\,\,2 \\ \end{align} \right]\,\,[2\text{   }1\text{   }-1]\]  = [MP PET 2000]
  A)
  [-1] done clear
  B)
  \[\left[ \begin{align}   & \,\,\,2 \\  & -1 \\  & -2 \\ \end{align} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    \,\,2 & \,\,1 & -1  \\    -2 & -1 & \,\,1  \\    \,\,4 & \,\,2 & -2  \\ \end{matrix} \right]\] done clear
  D)
  Not defined done clear
  View Solution play_arrow

• question_answer75)
  If two matrices A and B are of order p × q and r × s respectively, can be subtracted only, if [RPET 2000]
  A)
  \[p=q\] done clear
  B)
  \[p=q,r=s\] done clear
  C)
  \[p=r,q=s\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer76)
  If \[A=\left[ \begin{matrix}    1 & 3  \\    2 & 1  \\ \end{matrix} \right]\], then determinant of \[{{A}^{2}}-2A\]is [EAMCET 2000]
  A)
  5 done clear
  B)
  25 done clear
  C)
  -5 done clear
  D)
  -25 done clear
  View Solution play_arrow

• question_answer77)
  If the matrix \[AB=O\], then [Pb. CET 2000]
  A)
  \[A=O\] or \[B=O\] done clear
  B)
  \[A=O\] and \[B=O\] done clear
  C)
  It is not necessary that either \[A=O\]or \[B=O\] done clear
  D)
  \[A\ne O,B\ne O\] done clear
  View Solution play_arrow

• question_answer78)
  If \[{{a}_{ij}}=\frac{1}{2}(3i-2j)\]and \[A={{[{{a}_{ij}}]}_{2\times 2}}\], then A is equal to [RPET 2001]
  A)
  \[\left[ \begin{matrix}    1/2 & 2  \\    -1/2 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1/2 & -1/2  \\    2 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    2 & 2  \\    1/2 & -1/2  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer79)
  If \[2X-\left[ \begin{matrix}    1 & 2  \\    7 & 4  \\ \end{matrix} \right]=\left[ \begin{matrix}    3 & 2  \\    0 & -2  \\ \end{matrix} \right]\], then X is equal to   [RPET 2001]
  A)
  \[\left[ \begin{matrix}    2 & 2  \\    7 & 4  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & 2  \\    7/2 & 2  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    2 & 2  \\    7/2 & 1  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer80)
  If \[A=\left[ \begin{matrix}    1 & 2  \\    0 & 1  \\ \end{matrix} \right],\]then \[{{A}^{n}}=\] [Kerala (Engg.) 2001]
  A)
  \[\left[ \begin{matrix}    1 & 2n  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2 & n  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 2n  \\    0 & -1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 2n  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer81)
  If \[A=\left[ \begin{matrix}    0 & 2  \\    3 & -4  \\ \end{matrix} \right]\] and  \[kA=\left[ \begin{matrix}    0 & 3a  \\    2b & 24  \\ \end{matrix} \right]\], then the values of k, a, b are respectively [EAMCET 2001]
  A)
  \[-\,6,-\,12,-\,18\] done clear
  B)
  - 6, 4, 9 done clear
  C)
  \[-\,6,-\,4,-\,9\] done clear
  D)
  - 6, 12, 18 done clear
  View Solution play_arrow

• question_answer82)
  If \[\left[ \begin{matrix}    2+x & 3 & 4  \\    1 & -1 & 2  \\    x & 1 & -5  \\ \end{matrix} \right]\]is a singular matrix, then x is  [Kerala (Engg.) 2001]
  A)
  \[\frac{13}{25}\] done clear
  B)
  \[-\frac{25}{13}\] done clear
  C)
  \[\frac{5}{13}\] done clear
  D)
  \[\frac{25}{13}\] done clear
  View Solution play_arrow

• question_answer83)
  For the matrix \[A=\left[ \begin{matrix}    1 & 1 & 0  \\    1 & 2 & 1  \\    2 & 1 & 0  \\ \end{matrix} \right]\], which of the following is correct  [Kerala (Engg.)2001]
  A)
  \[{{A}^{3}}+3{{A}^{2}}-I=O\] done clear
  B)
  \[{{A}^{3}}-3{{A}^{2}}-I=O\] done clear
  C)
  \[{{A}^{3}}+2{{A}^{2}}-I=O\] done clear
  D)
  \[{{A}^{3}}-{{A}^{2}}+I=O\] done clear
  View Solution play_arrow

• question_answer84)
  The matrix \[\left[ \begin{matrix}    2 & \lambda  & -4  \\    -1 & 3 & 4  \\    1 & -2 & -3  \\ \end{matrix} \right]\]is non singular, if  [Kurukshetra CEE 2002]
  A)
  \[\lambda \ne -2\] done clear
  B)
  \[\lambda \ne 2\] done clear
  C)
  \[\lambda \ne 3\] done clear
  D)
  \[\lambda \ne -3\] done clear
  View Solution play_arrow

• question_answer85)
  If \[A=\left[ \begin{matrix}    1 & -1  \\    2 & -1  \\ \end{matrix} \right],\,\,B=\left[ \begin{matrix}    a & 1  \\    b & -1  \\ \end{matrix} \right]\] and \[{{(A+B)}^{2}}={{A}^{2}}+{{B}^{2}}\],  then the value of a and b are [Kurukshetra CEE 2002]
  A)
  \[a=4,b=1\] done clear
  B)
  \[a=1,b=4\] done clear
  C)
  \[a=0,b=4\] done clear
  D)
  \[a=2,b=4\] done clear
  View Solution play_arrow

• question_answer86)
  If \[A,B\]are square matrices of order 3, A is non- singular and \[AB=O\], then B is a [EAMCET 2002]
  A)
  Null matrix done clear
  B)
  Singular matrix done clear
  C)
  Unit matrix done clear
  D)
  Non- singular matrix done clear
  View Solution play_arrow

• question_answer87)
  If \[A=\left[ \begin{matrix}    3 & -5  \\    -4 & 2  \\ \end{matrix} \right],\]then \[{{A}^{2}}-5A=\] [RPET 2002]
  A)
  I done clear
  B)
  14 I done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer88)
  If matrix \[A=\left[ \begin{matrix}    0 & -1  \\    1 & 0  \\ \end{matrix} \right]\], then \[{{A}^{16}}=\][Karnataka CET 2002]
  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_answer89)
  If \[A=\left[ \begin{matrix}    1 & 1  \\    1 & 1  \\ \end{matrix} \right],\]then \[{{A}^{100}}=\] [UPSEAT 2002; MP PET 2004]
  A)
  \[{{2}^{100}}A\] done clear
  B)
  \[{{2}^{99}}A\] done clear
  C)
  \[{{2}^{101}}A\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer90)
  Which is true about matrix multiplication  [UPSEAT 2002]
  A)
  It is commutative done clear
  B)
  It is associative done clear
  C)
  Both (a) and (b) done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer91)
  Matrix \[A=\left[ \begin{matrix}    1 & 0 & -k  \\    2 & 1 & 3  \\    k & 0 & 1  \\ \end{matrix} \right]\]is invertible for  [UPSEAT 2002]
  A)
  \[k=1\] done clear
  B)
  \[k=-1\] done clear
  C)
  \[k=0\] done clear
  D)
  All real k done clear
  View Solution play_arrow

• question_answer92)
  If \[\left[ \begin{matrix}    x+y & 2x+z  \\    x-y & 2z+w  \\ \end{matrix} \right]=\left[ \begin{matrix}    4 & 7  \\    0 & 10  \\ \end{matrix} \right]\], then values of\[x,y,z,w\]are [RPET 2002]
  A)
  2, 2, 3, 4 done clear
  B)
  2, 3, 1, 2 done clear
  C)
  3, 3, 0, 1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer93)
  The value of a for which the matrix \[A=\left( \begin{matrix}    a & 2  \\    2 & 4  \\ \end{matrix} \right)\]is singular if [Kerala (Engg.)2002]
  A)
  \[a\ne 1\] done clear
  B)
  \[a=1\] done clear
  C)
  \[a=0\] done clear
  D)
  \[a=-1\] done clear
  View Solution play_arrow

• question_answer94)
  If \[A=\left( \begin{matrix}    2 & -1  \\    -1 & 2  \\ \end{matrix} \right)\]and I is the unit matrix of order 2, then \[{{A}^{2}}\] equals [Kerala (Engg.) 2002]
  A)
  \[4A-3I\] done clear
  B)
  \[3A-AI\] done clear
  C)
  \[A-I\] done clear
  D)
  \[A+I\] done clear
  View Solution play_arrow

• question_answer95)
  If \[P=\left( \begin{matrix}    i & 0 & -i  \\    0 & -i & i  \\    -i & i & 0  \\ \end{matrix} \right)\] and \[Q=\left( \begin{matrix}    -i & i  \\    0 & 0  \\    i & -i  \\ \end{matrix} \right)\],then \[PQ\] is equal to [Kerala (Engg.) 2002]
  A)
  \[\left( \begin{matrix}    -2 & 2  \\    1 & -1  \\    1 & -1  \\ \end{matrix} \right)\] done clear
  B)
  \[\left( \begin{matrix}    \,2 & -2  \\    -1 & \,\,\,1  \\    -1 & \,\,\,1  \\ \end{matrix} \right)\] done clear
  C)
  \[\left( \begin{matrix}    2 & -2  \\    -1 & \,\,1  \\ \end{matrix} \right)\] done clear
  D)
  \[\left( \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 0  \\    0 & 0 & 1  \\ \end{matrix} \right)\] done clear
  View Solution play_arrow

• question_answer96)
  If I  is a unit matrix of order 10, then the determinant of I is equal to [Kerala (Engg.) 2002]
  A)
  10 done clear
  B)
  1 done clear
  C)
  1/10 done clear
  D)
  9 done clear
  View Solution play_arrow

• question_answer97)
  Assuming that the sums and products given below are defined, which of the following is not true for matrices [Karnataka CET 2003]
  A)
  \[A+B=B+A\] done clear
  B)
  \[AB=AC\]does not imply \[B=C\] done clear
  C)
  \[AB=O\]implies \[A=O\]or \[B=O\] done clear
  D)
  \[(AB{)}'={B}'{A}'\] done clear
  View Solution play_arrow

• question_answer98)
  If \[A=\left[ \begin{matrix}    1 & 2 & -1  \\    3 & 0 & \,\,2  \\    4 & 5 & \,\,0  \\ \end{matrix} \right]\], \[B=\left[ \begin{matrix}    1 & 0 & 0  \\    2 & 1 & 0  \\    0 & 1 & 3  \\ \end{matrix} \right],\]then \[AB\] is [MP PET 2003]
  A)
  \[\left[ \begin{matrix}    5 & 1 & -3  \\    3 & 2 & 6  \\    14 & 5 & 0  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    11 & 4 & 3  \\    1 & 2 & 3  \\    0 & 3 & 3  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 8 & 4  \\    2 & 9 & 6  \\    0 & 2 & 0  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    0 & 1 & 2  \\    5 & 4 & 3  \\    1 & 8 & 2  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer99)
  If A and B are \[3\times 3\]matrices such that \[AB=A\] and \[BA=B\], then [Orissa JEE 2003]
  A)
  \[{{A}^{2}}=A\]and \[{{B}^{2}}\ne B\] done clear
  B)
  \[{{A}^{2}}\ne A\]and \[{{B}^{2}}=B\] done clear
  C)
  \[{{A}^{2}}=A\]and \[{{B}^{2}}=B\] done clear
  D)
  \[{{A}^{2}}\ne A\]and \[{{B}^{2}}\ne B\] done clear
  View Solution play_arrow

• question_answer100)
  If \[A=\left[ \begin{matrix}    \alpha  & 0  \\    1 & 1  \\ \end{matrix} \right]\]and \[B=\left[ \begin{matrix}    1 & 0  \\    5 & 1  \\ \end{matrix} \right]\], then value of\[\alpha \]for which \[{{A}^{2}}=B\], is  [IIT Screening 2003]
  A)
  1 done clear
  B)
  -1 done clear
  C)
  4 done clear
  D)
  No real values done clear
  View Solution play_arrow

• question_answer101)
  \[\left[ \begin{matrix}    7 & 1 & 2  \\    9 & 2 & 1  \\ \end{matrix} \right]\left[ \begin{matrix}    3  \\    4  \\    5  \\ \end{matrix} \right]+2\left[ \begin{matrix}    4  \\    2  \\ \end{matrix} \right]\]  is equal to [DCE 2002]
  A)
  \[\left[ \begin{align}   & 43 \\  & 44 \\ \end{align} \right]\] done clear
  B)
  \[\left[ \begin{align}   & 43 \\  & 45 \\ \end{align} \right]\] done clear
  C)
  \[\left[ \begin{align}   & 45 \\  & 44 \\ \end{align} \right]\] done clear
  D)
  \[\left[ \begin{align}   & 44 \\  & 45 \\ \end{align} \right]\] done clear
  View Solution play_arrow

• question_answer102)
  Let \[A=\left( \begin{matrix}    0 & 0 & -1  \\    0 & -1 & 0  \\    -1 & 0 & 0  \\ \end{matrix} \right)\], the only correct statement about the matrix A is [AIEEE 2004]
  A)
  \[{{A}^{2}}=I\] done clear
  B)
  \[A=(-1)\,I,\]where I is a unit matrix done clear
  C)
  \[{{A}^{-1}}\]does not exist done clear
  D)
  A is a zero matrix done clear
  View Solution play_arrow

• question_answer103)
  If \[A=\left( \begin{matrix}    1 & 2 & 3  \\    3 & 1 & 2  \\    2 & 3 & 1  \\ \end{matrix} \right)\] and \[B=\left( \begin{matrix}    -5 & 7 & 1  \\    1 & -5 & 7  \\    7 & 1 & -5  \\ \end{matrix} \right)\] then \[AB\] is equal to [Pb. CET 2002]
  A)
  \[{{I}_{3}}\] done clear
  B)
  \[2{{I}_{3}}\] done clear
  C)
  \[4{{I}_{3}}\] done clear
  D)
  \[18{{I}_{3}}\] done clear
  View Solution play_arrow

• question_answer104)
  What must be the matrix X if \[2X+\left[ \begin{matrix}    1 & 2  \\    3 & 4  \\ \end{matrix} \right]=\left[ \begin{matrix}    3 & 8  \\    7 & 2  \\ \end{matrix} \right]\] [Karnataka CET 2004]
  A)
  \[\left[ \begin{matrix}    1 & 3  \\    2 & -1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & -3  \\    2 & -1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    2 & 6  \\    4 & -2  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    2 & -6  \\    4 & -2  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer105)
   If \[A+B=\left[ \begin{matrix}    1 & 0  \\    1 & 1  \\ \end{matrix} \right]\]and \[A-2B=\left[ \begin{matrix}    -1 & 1  \\    0 & -1  \\ \end{matrix} \right]\,,\]then A= [Karnataka CET 1994]
  A)
  \[\left[ \begin{matrix}    1 & 1  \\    2 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2/3 & 1/3  \\    1/3 & 2/3  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1/3 & 1/3  \\    2/3 & 1/3  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer106)
  The identity element in the group \[M=\left\{ \left. \left( \begin{matrix}    x & x  \\    x & x  \\ \end{matrix} \right) \right|x\in R;\,x\ne 0\, \right\}\] with respect to matrix multiplication is [Karnataka CET 2005]
  A)
  \[\left( \begin{matrix}    1 & 1  \\    1 & 1  \\ \end{matrix} \right)\] done clear
  B)
  \[\frac{1}{2}\left( \begin{matrix}    1 & 1  \\    1 & 1  \\ \end{matrix} \right)\] done clear
  C)
  \[\left( \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right)\] done clear
  D)
  \[\left( \begin{matrix}    0 & 1  \\    1 & 0  \\ \end{matrix} \right)\] done clear
  View Solution play_arrow

• question_answer107)
  If \[A=\left[ \begin{matrix}    1 & 0  \\    1 & 1  \\ \end{matrix} \right]\] and \[I=\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\], then which one of the following holds for all \[n\ge 1\], (by the principal of mathematical induction) [AIEEE 2005]
  A)
  \[{{A}^{n}}=nA+(n-1)I\] done clear
  B)
  \[{{A}^{n}}={{2}^{n-1}}A+(n-1)I\] done clear
  C)
  \[{{A}^{n}}=nA-(n-1)I\] done clear
  D)
  \[{{A}^{n}}={{2}^{n-1}}A-(n-1)I\] done clear
  View Solution play_arrow