JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

done Special types of matrices, Transpose, Adjoint and Inverse of matrices Total Questions - 97

• question_answer1)
  Inverse of the matrix \[\left[ \begin{matrix}    3 & -2 & -1  \\    -4 & 1 & -1  \\    2 & 0 & 1  \\ \end{matrix} \right]\] is  [MP PET 1990]
  A)
  \[\left[ \begin{matrix}    1 & 2 & 3  \\    3 & 3 & 7  \\    -2 & -4 & -5  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & -3 & 5  \\    7 & 4 & 6  \\    4 & 2 & 7  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 2 & 3  \\    2 & 5 & 7  \\    -2 & -4 & -5  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & -3 & 5  \\    7 & 4 & 6  \\    4 & 2 & -7  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer2)
  If A  and B  are non-singular matrices, then  [MP PET 1991; Kurukshetra CEE 1998]
  A)
  \[{{(AB)}^{-1}}={{A}^{-1}}{{B}^{-1}}\] done clear
  B)
  \[AB=BA\] done clear
  C)
  \[(AB{)}'={A}'{B}'\] done clear
  D)
  \[{{(AB)}^{-1}}={{B}^{-1}}{{A}^{-1}}\] done clear
  View Solution play_arrow

• question_answer3)
  Adjoint of the matrix \[N=\left[ \begin{matrix}    -4 & -3 & -3  \\    1 & 0 & 1  \\    4 & 4 & 3  \\ \end{matrix} \right]\]is [MP PET 1989]
  A)
  N done clear
  B)
  2N done clear
  C)
  - N done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer4)
  From the following find the correct relation [MP PET 1990]
  A)
  \[(AB{)}'={A}'{B}'\] done clear
  B)
  \[(AB{)}'={B}'{A}'\] done clear
  C)
  \[{{A}^{-1}}=\frac{adj\,A}{A}\] done clear
  D)
  \[{{(AB)}^{-1}}={{A}^{-1}}{{B}^{-1}}\] done clear
  View Solution play_arrow

• question_answer5)
  If  A is involutory matrix and I  is unit matrix of same order, then \[(I-A)(I+A)\] is
  A)
  Zero matrix done clear
  B)
  A done clear
  C)
  I done clear
  D)
  2A done clear
  View Solution play_arrow

• question_answer6)
  If k  is a scalar and I  is a unit matrix of order 3, then \[adj(k\,I)=\]                              [MP PET 1991; Pb. CET 2003]
  A)
  \[{{k}^{3}}I\] done clear
  B)
  \[{{k}^{2}}I\] done clear
  C)
  \[-{{k}^{3}}I\] done clear
  D)
  \[-{{k}^{2}}I\] done clear
  View Solution play_arrow

• question_answer7)
  If A is a \[n\times n\]matrix, then adj(adj A)=
  A)
  \[|A|{{\,}^{n-1}}A\] done clear
  B)
  \[|A|{{\,}^{n-2}}A\] done clear
  C)
  \[|A{{|}^{n}}n\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer8)
  If \[A=\left[ \begin{matrix}    i & 0  \\    0 & i/2  \\ \end{matrix} \right]\]\[(i=\sqrt{-1}),\]then \[{{A}^{-1}}\]= [MP PET 1992]
  A)
  \[\left[ \begin{matrix}    i & 0  \\    0 & i/2  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    -i & 0  \\    0 & -2i  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    i & 0  \\    0 & 2i  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    0 & i  \\    2i & 0  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer9)
  If A  is a non- singular matrix, then A(adj A) =
  A)
  A done clear
  B)
  I done clear
  C)
  |A|I done clear
  D)
  \[|A{{|}^{2}}I\] done clear
  View Solution play_arrow

• question_answer10)
  The element of second row and third column in the inverse of  \[\left[ \begin{matrix}    1 & 2 & 1  \\    2 & 1 & 0  \\    -1 & 0 & 1  \\ \end{matrix} \right]\] is [MP PET 1992]
  A)
  - 2 done clear
  B)
  - 1 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer11)
  If \[R(t)=\left[ \begin{matrix}    \cos t & \sin t  \\    -\sin t & \cos t  \\ \end{matrix} \right],\]then \[R(s).\,R(t)=\] [Roorkee 1981]
  A)
  \[R(s)+R(t)\] done clear
  B)
  \[R\,(st)\] done clear
  C)
  \[R(s+t)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  If  A  and B  be symmetric matrices of the same order, then \[AB-BA\] will be a
  A)
  Symmetric matrix done clear
  B)
  Skew symmetric matrix done clear
  C)
  Null matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  If A is a symmetric matrix, then matrix \[{M}'AM\]is [MP PET 1990]
  A)
  Symmetric done clear
  B)
  Skew-symmetric done clear
  C)
  Hermitian done clear
  D)
  Skew-Hermitian done clear
  View Solution play_arrow

• question_answer14)
  An orthogonal matrix is
  A)
  \[\left[ \begin{matrix}    \cos \alpha  & 2\sin \alpha   \\    -2\sin \alpha  & \cos \alpha   \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    \cos \alpha  & \sin \alpha   \\    -\sin \alpha  & \cos \alpha   \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    \cos \alpha  & \sin \alpha   \\    \sin \alpha  & \cos \alpha   \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 1  \\    1 & 1  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer15)
  If \[A=\left[ \begin{matrix}    a & c  \\    d & b  \\ \end{matrix} \right],\]then \[{{A}^{-1}}\]= [MP PET 1988]
  A)
  \[\frac{1}{ab-cd}\left[ \begin{matrix}    b & -c  \\    -d & a  \\ \end{matrix} \right]\] done clear
  B)
  \[\frac{1}{ad-bc}\left[ \begin{matrix}    b & -c  \\    -d & a  \\ \end{matrix} \right]\] done clear
  C)
  \[\frac{1}{ab-cd}\left[ \begin{matrix}    b & d  \\    c & a  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer16)
  The inverse of the matrix \[\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 0  \\    0 & 0 & 1  \\ \end{matrix} \right]\] is  [MP PET 1989; Pb. CET 1989, 1993]
  A)
  \[\left[ \begin{matrix}    0 & 0 & 1  \\    0 & 1 & 0  \\    1 & 0 & 0  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 0  \\    0 & 0 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    0 & 1 & 0  \\    0 & 0 & 1  \\    1 & 0 & 0  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 0 & 1  \\    0 & 1 & 0  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer17)
  The inverse of \[\left[ \begin{matrix}    2 & -3  \\    -4 & 2  \\ \end{matrix} \right]\]is   [MP PET 1993; Pb. CET 2000]
  A)
  \[\frac{-1}{8}\,\left[ \begin{matrix}    2 & 3  \\    4 & 2  \\ \end{matrix} \right]\] done clear
  B)
  \[\frac{-1}{8}\,\left[ \begin{matrix}    3 & 2  \\    2 & 4  \\ \end{matrix} \right]\] done clear
  C)
  \[\frac{1}{8}\,\left[ \begin{matrix}    2 & 3  \\    4 & 2  \\ \end{matrix} \right]\] done clear
  D)
  \[\frac{1}{8}\,\left[ \begin{matrix}    3 & 2  \\    2 & 4  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer18)
  Let \[A=\left[ \begin{matrix}    1 & 0 & 0  \\    5 & 2 & 0  \\    -1 & 6 & 1  \\ \end{matrix} \right]\], then the adjoint of A is  [MNR 1982]
  A)
  \[\left[ \begin{matrix}    2 & -5 & 32  \\    0 & 1 & -6  \\    0 & 0 & 2  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    -1 & 0 & 0  \\    -5 & -2 & 0  \\    1 & -6 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -1 & 0 & 0  \\    -5 & -2 & 0  \\    1 & -6 & -1  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  If \[A=\left[ \begin{matrix}    3 & 2  \\    1 & 4  \\ \end{matrix} \right]\], then \[A(adj\,A)=\]  [MP PET 1995; RPET 1997]
  A)
  \[\left[ \begin{matrix}    10 & 0  \\    0 & 10  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    0 & 10  \\    10 & 0  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    10 & 1  \\    1 & 10  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  If A is a square matrix, then which of the following matrices is not symmetric 
  A)
  \[A+{A}'\] done clear
  B)
  \[A{A}'\] done clear
  C)
  \[{A}'A\] done clear
  D)
  \[A-{A}'\] done clear
  View Solution play_arrow

• question_answer21)
  If \[A=\left[ \begin{matrix}    \cos \alpha  & \sin \alpha   \\    -\sin \alpha  & \cos \alpha   \\ \end{matrix} \right]\] and \[A\,\,adj\]\[A=\left[ \begin{matrix}    k & 0  \\    0 & k  \\ \end{matrix} \right],\] then k  is equal to                                [MP PET 1993; Pb. CET 2001]
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[\sin \alpha \cos \alpha \] done clear
  D)
  \[\cos 2\alpha \] done clear
  View Solution play_arrow

• question_answer22)
  If a matrix A is such that \[3{{A}^{3}}+2{{A}^{2}}+5A+I=0,\] then its inverse is
  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)
  None of these done clear
  View Solution play_arrow

• question_answer23)
  If A and B are square matrices of the same order, then   [Pb. CET 1992; Roorkee 1995]
  A)
  \[(AB{)}'={A}'{B}'\] done clear
  B)
  \[(AB{)}'={B}'{A}'\] done clear
  C)
  \[AB=O;\]If \[|A|\,=0\]or \[|B|\,=0\] done clear
  D)
  \[AB=O;\text{If}\,\,A=I\,\,\text{or }B=I\] done clear
  View Solution play_arrow

• question_answer24)
  Which of the following (i) Adjoint of a symmetric matrix is symmetric, (ii) Adjoint of a unit matrix is a unit matrix, (iii) \[A(adj\,A)=(adj\,A)\]\[A=\,|A|I\]and (iv) Adjoint of a diagonal matrix is a diagonal matrix, is/are incorrect
  A)
  (i) done clear
  B)
  (ii) done clear
  C)
  (iii) and (iv) done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer25)
  \[{{\left[ \begin{matrix}    1 & 3  \\    3 & 10  \\ \end{matrix} \right]}^{-1}}=\]   [EAMCET 1994; DCE 1999]
  A)
  \[\left[ \begin{matrix}    10 & 3  \\    3 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    10 & -3  \\    -3 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 3  \\    3 & 10  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    -1 & -3  \\    -3 & -10  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer26)
  The inverse of  a symmetric matrix is
  A)
  Symmetric done clear
  B)
  Skew symmetric done clear
  C)
  Diagonal matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer27)
  If A is a symmetric matrix and \[n\in N\], then \[{{A}^{n}}\]is
  A)
  Symmetric done clear
  B)
  Skew symmetric done clear
  C)
  A Diagonal matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer28)
  If A is a skew symmetric matrix and n is a positive integer, then \[{{A}^{n}}\]is
  A)
  A symmetric matrix done clear
  B)
  Skew-symmetric matrix done clear
  C)
  Diagonal matrix done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer29)
  \[{{\left[ \begin{matrix}    -6 & 5  \\    -7 & 6  \\ \end{matrix} \right]}^{-1}}\]= [Karnataka CET 1994]
  A)
  \[\left[ \begin{matrix}    -6 & 5  \\    -7 & 6  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    6 & -5  \\    -7 & 6  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    6 & 5  \\    7 & 6  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    6 & -5  \\    7 & -6  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer30)
  The adjoint of \[\left[ \begin{matrix}    1 & 1 & 1  \\    1 & 2 & -3  \\    2 & -1 & 3  \\ \end{matrix} \right]\]is [RPET 1993]
  A)
  \[\left[ \begin{matrix}    3 & -9 & -5  \\    -4 & 1 & 3  \\    -5 & 4 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    3 & -4 & -5  \\    -9 & 1 & 4  \\    -5 & 3 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -3 & \,\,4 & 5  \\    9 & -1 & -4  \\    5 & -3 & -1  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer31)
  If \[A=\left[ \begin{matrix}    5 & 2  \\    3 & 1  \\ \end{matrix} \right],\]then \[{{A}^{-1}}\]= [EAMCET 1988]
  A)
  \[\left[ \begin{matrix}    1 & -2  \\    -3 & 5  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    -1 & 2  \\    3 & -5  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -1 & -2  \\    -3 & -5  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 2  \\    3 & 5  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer32)
  If \[A=\left[ \begin{matrix}    4 & x+2  \\    2x-3 & x+1  \\ \end{matrix} \right]\]is symmetric, then x = [Karnataka CET 1994]
  A)
  3 done clear
  B)
  5 done clear
  C)
  2 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer33)
  The inverse of the matrix \[\left[ \begin{matrix}    3 & -2  \\    1 & 4  \\ \end{matrix} \right]\]is  [MP PET 1994]
  A)
  \[\left[ \begin{matrix}    \frac{4}{14} & \frac{2}{14}  \\    \frac{-1}{14} & \frac{3}{14}  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    \frac{3}{14} & \frac{-2}{14}  \\    \frac{1}{14} & \frac{4}{14}  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    \frac{4}{14} & \frac{-2}{14}  \\    \frac{1}{14} & \frac{3}{14}  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    \frac{3}{14} & \frac{2}{14}  \\    \frac{1}{14} & \frac{4}{14}  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer34)
  Matrix \[\left[ \begin{matrix}    0 & -4 & 1  \\    4 & 0 & -5  \\    -1 & 5 & 0  \\ \end{matrix} \right]\]is
  A)
  Orthogonal done clear
  B)
  Idempotent done clear
  C)
  Skew- symmetric done clear
  D)
  Symmetric done clear
  View Solution play_arrow

• question_answer35)
  If \[A=\left[ \begin{matrix}    1 & -2  \\    5 & 3  \\ \end{matrix} \right]\], then \[A+{{A}^{T}}\]equals [RPET 1994]
  A)
  \[\left[ \begin{matrix}    2 & 3  \\    3 & 6  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2 & -4  \\    10 & 6  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    2 & 4  \\    -10 & 6  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer36)
  The inverse of matrix \[A=\left[ \begin{matrix}    0 & 1 & 0  \\    1 & 0 & 0  \\    0 & 0 & 1  \\ \end{matrix} \right]\]is  [Karnataka CET 1993]
  A)
  A done clear
  B)
  \[{{A}^{T}}\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 0  \\    0 & 0 & 1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 0 & 0  \\    1 & 0 & 0  \\    0 & 1 & 0  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer37)
   If A is a singular matrix, then adj A is  [Karnataka CET 1993]
  A)
  Singular done clear
  B)
  Non-singular done clear
  C)
  Symmetric done clear
  D)
  Not defined done clear
  View Solution play_arrow

• question_answer38)
  The inverse of  \[\left[ \begin{matrix}    1 & 2 & 3  \\    0 & 1 & 2  \\    0 & 0 & 1  \\ \end{matrix} \right]\]is [EAMCET 1990]
  A)
  \[\left[ \begin{matrix}    1 & -2 & 1  \\    0 & 1 & -2  \\    0 & 0 & 0  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & -2 & 1  \\    0 & 1 & -2  \\    0 & 0 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 2 & 1  \\    0 & 1 & 2  \\    0 & 0 & 1  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer39)
  Which one of the following statements is true [MP PET 1996]
  A)
  Non- singular square matrix does not have a unique inverse done clear
  B)
  Determinant of a non-singular matrix is zero done clear
  C)
  If \[{A}'=A,\]then A is a square matrix done clear
  D)
  If \[|A|\,\ne 0\], then \[|A.adj\,A|\,=\,|A{{|}^{(n-1)}}\], where \[A={{[{{a}_{ij}}]}_{n\times n}}\] done clear
  View Solution play_arrow

• question_answer40)
  If matrix \[A=\left[ \begin{matrix}    1 & -1  \\    1 & 1  \\ \end{matrix} \right],\]then [MP PET 1996]
  A)
  \[{A}'=\left[ \begin{matrix}    1 & \,\,1  \\    1 & -1  \\ \end{matrix} \right]\] done clear
  B)
  \[{{A}^{-1}}=\left[ \begin{matrix}    \,\,1 & 1  \\    -1 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[A.\,\,\left[ \begin{matrix}    \,\,1 & 1  \\    -1 & 1  \\ \end{matrix} \right]=2I\] done clear
  D)
  \[\lambda A=\left[ \begin{matrix}    \lambda  & -\lambda   \\    1 & -1  \\ \end{matrix} \right]\]where \[\lambda \]is a non zero scalar done clear
  View Solution play_arrow

• question_answer41)
  Which of the following is not true   [Kurukshetra CEE 1996]
  A)
  Every skew-symmetric matrix of odd order is non-singular done clear
  B)
  If determinant of a square matrix is non-zero, then it is non singular done clear
  C)
  Adjoint of symmetric matrix is symmetric done clear
  D)
  Adjoint of a diagonal matrix is diagonal done clear
  View Solution play_arrow

• question_answer42)
  If \[A=\left( \begin{matrix}    1 & 2 & 0  \\    0 & 1 & 2  \\    2 & 0 & 1  \\ \end{matrix} \right),\]then adj A [RPET 1996]
  A)
  \[\left( \begin{matrix}    1 & 4 & -2  \\    -2 & 1 & 4  \\    4 & -2 & 1  \\ \end{matrix} \right)\] done clear
  B)
  \[\left( \begin{matrix}    1 & -2 & 4  \\    4 & 1 & -2  \\    -2 & 4 & 1  \\ \end{matrix} \right)\] done clear
  C)
  \[\left( \begin{matrix}    1 & 2 & 4  \\    -4 & 1 & 2  \\    -4 & -2 & 1  \\ \end{matrix} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer43)
  Which one of the following is correct  [Kurukshetra CEE 1998]
  A)
  Skew- symmetric matrix of odd order is non-singular done clear
  B)
  Skew-symmetric matrix of odd order is singular done clear
  C)
  Skew-symmetric matrix of even order is always singular done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer44)
  \[Adj.\]\[(AB)-(Adj.\,B)(Adj.\,A)=\] [MP PET 1997]
  A)
  \[Adj.A-Adj\,B\] done clear
  B)
  I done clear
  C)
  O done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer45)
  If \[A=\left( \begin{matrix}    3 & 2  \\    0 & 1  \\ \end{matrix} \right)\], then \[{{({{A}^{-1}})}^{3}}\]is equal to [MP PET 1997; Pb. CET 2003]
  A)
  \[\frac{1}{27}\left( \begin{matrix}    1 & -26  \\    0 & 27  \\ \end{matrix} \right)\] done clear
  B)
  \[\frac{1}{27}\left( \begin{matrix}    -1 & 26  \\    0 & 27  \\ \end{matrix} \right)\] done clear
  C)
  \[\frac{1}{27}\left( \begin{matrix}    1 & -26  \\    0 & -27  \\ \end{matrix} \right)\] done clear
  D)
  \[\frac{1}{27}\left( \begin{matrix}    -1 & -26  \\    0 & -27  \\ \end{matrix} \right)\] done clear
  View Solution play_arrow

• question_answer46)
  The matrix \[\left( \begin{matrix}    1 & a & 2  \\    1 & 2 & 5  \\    2 & 1 & 1  \\ \end{matrix} \right)\]is not invertible, if ?a? has the value [MP PET 1998]
  A)
  2 done clear
  B)
  1 done clear
  C)
  0 done clear
  D)
  -1 done clear
  View Solution play_arrow

• question_answer47)
  For any \[2\times 2\] matrix A, if \[A(adj.\,\,A)\]= \[\left[ \begin{matrix}    10 & 0  \\    0 & 10  \\ \end{matrix} \right]\], then \[|A|\,=\] [MP PET 1999]
  A)
  0 done clear
  B)
  10 done clear
  C)
  20 done clear
  D)
  100 done clear
  View Solution play_arrow

• question_answer48)
  If A is a square matrix for which \[{{a}_{ij}}={{i}^{2}}-{{j}^{2}}\], then  A is   [RPET 1999]
  A)
  Zero matrix done clear
  B)
  Unit matrix done clear
  C)
  Symmetric matrix done clear
  D)
  Skew symmetric matrix done clear
  View Solution play_arrow

• question_answer49)
  Inverse matrix of \[\left[ \begin{matrix}    4 & 7  \\    1 & 2  \\ \end{matrix} \right]\]      [RPET 1996, 2001]
  A)
  \[\left[ \begin{matrix}    2 & -7  \\    -1 & 4  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2 & -1  \\    -7 & 4  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -2 & 7  \\    1 & -4  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    -2 & 1  \\    7 & -4  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer50)
  If \[A=\left[ \begin{matrix}    3 & -3 & 4  \\    2 & -3 & 4  \\    0 & -1 & 1  \\ \end{matrix} \right]\], then \[{{A}^{-1}}\]= [DCE 1999]
  A)
  A done clear
  B)
  \[{{A}^{2}}\] done clear
  C)
  \[{{A}^{3}}\] done clear
  D)
  \[{{A}^{4}}\] done clear
  View Solution play_arrow

• question_answer51)
  If  d is the determinant of a square matrix A of order n, then the determinant of its adjoint is  [EAMCET 2000]
  A)
  \[{{d}^{n}}\] done clear
  B)
  \[{{d}^{n-1}}\] done clear
  C)
  \[{{d}^{n+1}}\] done clear
  D)
  \[d\] done clear
  View Solution play_arrow

• question_answer52)
  If A and B are non-singular square matrices of same order, then \[adj(AB)\]is equal to [AMU 1999]
  A)
  \[(adj\,A)(adj\,B)\] done clear
  B)
  \[(adj\,B)(adj\,A)\] done clear
  C)
  \[(adj\,{{B}^{-1}})(adj\,{{A}^{-1}})\] done clear
  D)
  \[(adj\,{{A}^{-1}})(adj\,{{B}^{-1}})\] done clear
  View Solution play_arrow

• question_answer53)
  The element in the first row and third column of the inverse of the matrix \[\left[ \begin{matrix}    1 & 2 & -3  \\    0 & 1 & 2  \\    0 & 0 & 1  \\ \end{matrix} \right]\]is [MP PET 2000]
  A)
  - 2 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  7 done clear
  View Solution play_arrow

• question_answer54)
  For any square matrix A, \[A{{A}^{T}}\]is a [RPET 2000]
  A)
  Unit matrix done clear
  B)
  Symmetric matrix done clear
  C)
  Skew symmetric matrix done clear
  D)
  Diagonal matrix done clear
  View Solution play_arrow

• question_answer55)
  The matrix \[\left[ \begin{matrix}    \lambda  & -1 & 4  \\    -3 & 0 & 1  \\    -1 & 1 & 2  \\ \end{matrix} \right]\]is invertible, if [Kurukshetra CEE 1996]
  A)
  \[\lambda \ne -15\] done clear
  B)
  \[\lambda \ne -17\] done clear
  C)
  \[\lambda \ne -16\] done clear
  D)
  \[\lambda \ne -18\] done clear
  View Solution play_arrow

• question_answer56)
  For a matrix A, AI = A and \[A{{A}^{T}}=I\]is true for [RPET 2000]
  A)
  If A is a square matrix done clear
  B)
  If A is a non singular matrix done clear
  C)
  If A is a symmetric matrix done clear
  D)
  If A is any matrix done clear
  View Solution play_arrow

• question_answer57)
  If \[{{I}_{3}}\] is the identity matrix of order 3, then \[I_{3}^{-1}\]is [Pb. CET 2000]
  A)
  0 done clear
  B)
  \[3{{I}_{3}}\] done clear
  C)
  \[{{I}_{3}}\] done clear
  D)
  Does not exist done clear
  View Solution play_arrow

• question_answer58)
  The matrix \[\left[ \begin{matrix}    0 & 5 & -7  \\    -5 & 0 & 11  \\    7 & -11 & 0  \\ \end{matrix} \right]\]is known as  [Karnataka CET 2000]
  A)
  Upper triangular matrix done clear
  B)
  Skew symmetric matrix done clear
  C)
  Symmetric matrix done clear
  D)
  Diagonal matrix done clear
  View Solution play_arrow

• question_answer59)
  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 \[{{(AB)}^{T}}\]is equal to [RPET 2001]
  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)
  \[\left( \begin{matrix}    -3 & 7  \\    10 & 2  \\ \end{matrix} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer60)
  If \[A=\left[ \begin{matrix}    1 & -1  \\    2 & 3  \\ \end{matrix} \right]\], then adj A is equal to [RPET 2001]
  A)
  \[\left[ \begin{matrix}    -3 & -1  \\    2 & -1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    3 & 1  \\    -2 & 1  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    3 & -2  \\    1 & 1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    \,3 & -1  \\    -2 & 1  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer61)
  If A is a square matrix, then \[A+{{A}^{T}}\]is  [RPET 2001]
  A)
  Non singular matrix done clear
  B)
  Symmetric matrix done clear
  C)
  Skew-symmetric matrix done clear
  D)
  Unit matrix done clear
  View Solution play_arrow

• question_answer62)
  The inverse of a matrix \[A=\left( \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right)\]is  [AMU 2001]
  A)
  \[\left( \begin{matrix}    d & -b  \\    -c & a  \\ \end{matrix} \right)\] done clear
  B)
  \[\frac{1}{(ad-bc)}\left( \begin{matrix}    d & -b  \\    -c & a  \\ \end{matrix} \right)\] done clear
  C)
  \[\frac{1}{|A|}\left( \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right)\] done clear
  D)
  \[\left( \begin{matrix}    b & -a  \\    d & -c  \\ \end{matrix} \right)\] done clear
  View Solution play_arrow

• question_answer63)
  If \[A=\left[ \begin{matrix}    \cos \theta  & -\sin \theta   \\    \sin \theta  & \cos \theta   \\ \end{matrix} \right]\], then which of the following statements is not correct [DCE 2001]
  A)
  A is orthogonal matrix done clear
  B)
  \[{A}'\]is orthogonal matrix done clear
  C)
  Determinant A = 1 done clear
  D)
  A is not invertible done clear
  View Solution play_arrow

• question_answer64)
  If \[{{A}^{2}}-A+I=0\], then \[{{A}^{-1}}\]= [Kerala (Engg,) 2001; AIEEE 2005]
  A)
  \[{{A}^{-2}}\] done clear
  B)
  \[A+I\] done clear
  C)
  \[I-A\] done clear
  D)
  \[A-I\] done clear
  View Solution play_arrow

• question_answer65)
  If \[A=\left[ \begin{matrix}    2 & 2  \\    -3 & 2  \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix}    0 & -1  \\    1 & 0  \\ \end{matrix} \right],\] then \[{{({{B}^{-1}}{{A}^{-1}})}^{-1}}\]=  [EAMCET 2001]
  A)
  \[\left[ \begin{matrix}    2 & -2  \\    2 & 3  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    3 & -2  \\    2 & 2  \\ \end{matrix} \right]\] done clear
  C)
  \[\frac{1}{10}\left[ \begin{matrix}    2 & 2  \\    -2 & 3  \\ \end{matrix} \right]\] done clear
  D)
  \[\frac{1}{10}\left[ \begin{matrix}    3 & 2  \\    -2 & 2  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer66)
  A square matrix \[A=[{{a}_{ij}}]\] in which \[{{a}_{ij}}=0\] for \[i\,\ne j\] and \[{{a}_{ij}}=k\] (constant)  for \[i=j\] is called a  [EAMCET 2001]
  A)
  Unit matrix done clear
  B)
  Scalar matrix done clear
  C)
  Null matrix done clear
  D)
  Diagonal matrix done clear
  View Solution play_arrow

• question_answer67)
  For two invertible matrices A and B  of suitable orders, the  value of \[{{(AB)}^{-1}}\]is [Pb. CET 2000, RPET 2000, 02; Karnataka CET 2001]
  A)
  \[{{(BA)}^{-1}}\] done clear
  B)
  \[{{B}^{-1}}{{A}^{-1}}\] done clear
  C)
  \[{{A}^{-1}}{{B}^{-1}}\] done clear
  D)
  \[{{(A{B}')}^{-1}}\] done clear
  View Solution play_arrow

• question_answer68)
  If \[A=\left[ \begin{matrix}    -1 & 2  \\    2 & -1  \\ \end{matrix} \right]\]and \[B=\left[ \begin{align}   & 3 \\  & 1 \\ \end{align} \right],AX=B\], then \[X=\] [MP PET 2002]
  A)
  [5 7] done clear
  B)
  \[\frac{1}{3}\left[ \begin{align}   & 5 \\  & 7 \\ \end{align} \right]\] done clear
  C)
  \[\frac{1}{3}[5\,\,7]\] done clear
  D)
  \[\left[ \begin{align}   & 5 \\  & 7 \\ \end{align} \right]\] done clear
  View Solution play_arrow

• question_answer69)
  If  \[A=\left[ \begin{matrix}    1 & 2  \\    3 & -5  \\ \end{matrix} \right]\], then \[{{A}^{-1}}\]= [MP PET 2002]
  A)
  \[\left[ \begin{matrix}    -5 & -2  \\    -3 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    \frac{5}{11} & \frac{2}{11}  \\    \frac{3}{11} & -\frac{1}{11}  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -\frac{5}{11} & -\frac{2}{11}  \\    -\frac{3}{11} & -\frac{1}{11}  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    5 & 2  \\    3 & -1  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer70)
  If \[A=\left[ \begin{matrix}    2 & 3  \\    4 & 6  \\ \end{matrix} \right]\], then \[{{A}^{-1}}\]= [Karnataka CET 2001]
  A)
  \[\left[ \begin{matrix}    1 & 2  \\    -3/2 & 3  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2 & -3  \\    4 & 6  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    -2 & 4  \\    -3 & 6  \\ \end{matrix} \right]\] done clear
  D)
  Does not exist done clear
  View Solution play_arrow

• question_answer71)
  If \[A=\left[ \begin{matrix}    4 & 2  \\    3 & 4  \\ \end{matrix} \right]\],then \[|adj\,\,A|\]is equal to [UPSEAT 2003]
  A)
  16 done clear
  B)
  10 done clear
  C)
  6 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer72)
  The matrix \[A=\left[ \begin{matrix}    1 & -3 & -4  \\    -1 & \,\,\,3 & \,\,4  \\    1 & -3 & -4  \\ \end{matrix} \right]\] is nilpotent of index  [Kurukshetra CEE 2002]
  A)
  2 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer73)
  The matrix \[A=\left[ \begin{matrix}    i & 1-2i  \\    -1-2i & 0  \\ \end{matrix} \right]\]is which of the following   [Kurukshetra CEE 2002]
  A)
  Symmetric done clear
  B)
  Skew-symmetric done clear
  C)
  Hermitian done clear
  D)
  Skew-hermitian done clear
  View Solution play_arrow

• question_answer74)
  The adjoint matrix of \[\left[ \begin{matrix}    3 & -3 & 4  \\    2 & -3 & 4  \\    0 & -1 & 1  \\ \end{matrix} \right]\]is [MP PET 2003]
  A)
  \[\left[ \begin{matrix}    4 & 8 & 3  \\    2 & 1 & 6  \\    0 & 2 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & -1 & 0  \\    -2 & 3 & -4  \\    -2 & 3 & -3  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    11 & 9 & 3  \\    1 & 2 & 8  \\    6 & 9 & 1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & -2 & 1  \\    -1 & 3 & 3  \\    -2 & 3 & -3  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer75)
  \[A=\left[ \begin{matrix}    0 & 3  \\    2 & 0  \\ \end{matrix} \right]\]and \[{{A}^{-1}}=\lambda (adj(A)),\]then \[\lambda =\] [UPSEAT 2002]
  A)
  \[\frac{-1}{6}\] done clear
  B)
  \[\frac{1}{3}\] done clear
  C)
  \[\frac{-1}{3}\] done clear
  D)
  \[\frac{1}{6}\] done clear
  View Solution play_arrow

• question_answer76)
  Which of the following is true for matrix \[AB\] [RPET 2003]
  A)
  \[{{(AB)}^{-1}}={{A}^{-1}}{{B}^{-1}}\] done clear
  B)
  \[{{(AB)}^{-1}}={{B}^{-1}}{{A}^{-1}}\] done clear
  C)
  \[AB=BA\] done clear
  D)
  All of these done clear
  View Solution play_arrow

• question_answer77)
  If \[A=\left[ \begin{matrix}    \cos x & \sin x  \\    -\sin x & \cos x  \\ \end{matrix} \right]\], then \[A.\]\[(adj(A))\]= [RPET  2003]
  A)
  \[\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    0 & 1  \\    1 & 0  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 1  \\    0 & 0  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    -2 & 0  \\    0 & -2  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer78)
  The inverse matrix of \[\left[ \begin{matrix}    0 & 1 & 2  \\    1 & 2 & 3  \\    3 & 1 & 1  \\ \end{matrix} \right],\]is  [MP PET 2003]
  A)
  \[\left[ \begin{matrix}    \frac{1}{2} & -\frac{1}{2} & \frac{1}{2}  \\    -4 & 3 & -1  \\    \frac{5}{2} & \frac{-3}{2} & \frac{1}{2}  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    \frac{1}{2} & -4 & \frac{5}{2}  \\    1 & -6 & 3  \\    1 & 2 & -1  \\ \end{matrix} \right]\] done clear
  C)
  \[\frac{1}{2}\left[ \begin{matrix}    1 & 2 & 3  \\    3 & 2 & 1  \\    4 & 2 & 3  \\ \end{matrix} \right]\] done clear
  D)
  \[\frac{1}{2}\left[ \begin{matrix}    1 & -1 & -1  \\    -8 & 6 & -2  \\    5 & -3 & 1  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer79)
  The multiplicative inverse of matrix \[\left[ \begin{matrix}    2 & 1  \\    7 & 4  \\ \end{matrix} \right]\]is  [DCE 2002]
  A)
  \[\left[ \begin{matrix}    4 & -1  \\    -7 & -2  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    -4 & -1  \\    7 & -2  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    4 & -7  \\    7 & 2  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    4 & -1  \\    -7 & 2  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer80)
  If \[A=\left[ \begin{matrix}    -2 & 6  \\    -5 & 7  \\ \end{matrix} \right]\], then adj [UPSEAT 2002]
  A)
  \[\left[ \begin{matrix}    7 & -6  \\    5 & -2  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2 & -6  \\    5 & -7  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    7 & -5  \\    6 & -2  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer81)
  If matrix \[A=\left[ \begin{matrix}    3 & 2 & 4  \\    1 & 2 & -1  \\    0 & 1 & 1  \\ \end{matrix} \right]\]and \[{{A}^{-1}}=\frac{1}{K}adj(A),\] then \[K\]is  [UPSEAT 2002]
  A)
  7 done clear
  B)
  -7 done clear
  C)
  \[\frac{1}{7}\] done clear
  D)
  11 done clear
  View Solution play_arrow

• question_answer82)
  If \[A=\left[ \begin{matrix}    3 & 4  \\    5 & 7  \\ \end{matrix} \right]\], then \[A\,(adj\,A)\]= [RPET 2002]
  A)
  I done clear
  B)
  |A| done clear
  C)
  \[|A|I\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer83)
  If for the matrix A, \[{{A}^{3}}=I\], then \[{{A}^{-1}}=\] [RPET 2002]
  A)
  \[{{A}^{2}}\] done clear
  B)
  \[{{A}^{3}}\] done clear
  C)
  A done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer84)
  If \[A=\left[ \begin{matrix}    0 & 0 & 1  \\    0 & 1 & 0  \\    1 & 0 & 0  \\ \end{matrix} \right],\]then \[{{A}^{-1}}=\] [MP PET 2004]
  A)
  I done clear
  B)
  - I done clear
  C)
  - A done clear
  D)
  A done clear
  View Solution play_arrow

• question_answer85)
  Inverse of the matrix \[\left( \begin{matrix}    1 & -2  \\    3 & 4  \\ \end{matrix} \right)\] is [Karnataka CET 2003]
  A)
  \[\frac{1}{10}\left( \begin{matrix}    4 & 2  \\    -3 & 1  \\ \end{matrix} \right)\] done clear
  B)
  \[\frac{1}{10}\left( \begin{matrix}    1 & -2  \\    3 & 4  \\ \end{matrix} \right)\] done clear
  C)
  \[\frac{1}{10}\left( \begin{matrix}    4 & 2  \\    -3 & 1  \\ \end{matrix} \right)\] done clear
  D)
  \[\left( \begin{matrix}    4 & 2  \\    -3 & 1  \\ \end{matrix} \right)\] done clear
  View Solution play_arrow

• question_answer86)
  Inverse of the matrix \[\left[ \begin{matrix}    \cos 2\theta  & -\sin 2\theta   \\    \sin 2\theta  & \cos 2\theta   \\ \end{matrix} \right]\]is  [Karnataka CET 2004]
  A)
  \[\left[ \begin{matrix}    \cos 2\theta  & -\sin 2\theta   \\    \sin 2\theta  & \cos 2\theta   \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    \cos 2\theta  & \sin 2\theta   \\    \sin 2\theta  & -\cos 2\theta   \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    \cos 2\theta  & \sin 2\theta   \\    \sin 2\theta  & \cos 2\theta   \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    \cos 2\theta  & \sin 2\theta   \\    -\sin 2\theta  & \cos 2\theta   \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer87)
  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 [AIEEE 2004]
  A)
  5 done clear
  B)
    - 1 done clear
  C)
  2 done clear
  D)
  - 2 done clear
  View Solution play_arrow

• question_answer88)
  For any \[2\times 2\] matrix A, if \[A(adj\,A)=\left[ \begin{matrix}    10 & 0  \\    0 & 10  \\ \end{matrix} \right]\] then \[|A|\] is equal [Pb. CET 2002]
  A)
  0 done clear
  B)
  10 done clear
  C)
  20 done clear
  D)
  100 done clear
  View Solution play_arrow

• question_answer89)
  If \[A,B,C\]are three \[n\times n\]matrices, then \[(ABC{)}'=\] [MP PET 1988]
  A)
  \[{A}'\,{B}'\,{C}'\] done clear
  B)
  \[{C}'\,{B}'\,{A}'\] done clear
  C)
  \[{B}'\,{C}'\,{A}'\] done clear
  D)
  \[{B}'\,{A}'\,{C}'\] done clear
  View Solution play_arrow

• question_answer90)
  If X is a square matrix of order 3 × 3 and \[\lambda \] is a scalar, then adj (\[\lambda X)\] is equal to [J & K 2005]
  A)
  \[\lambda \,adjX\] done clear
  B)
  \[{{\lambda }^{3}}adj\,X\] done clear
  C)
  \[{{\lambda }^{2}}adj\,X\] done clear
  D)
  \[{{\lambda }^{4}}adj\,X\] done clear
  View Solution play_arrow

• question_answer91)
  If \[X=\left[ \begin{matrix}    -x & -y  \\    z & t  \\ \end{matrix} \right]\] then transpose of adj X is  [J & K 2005]
  A)
  \[\left[ \begin{matrix}    t & z  \\    -y & -x  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    t & y  \\    -z & -x  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    t & -z  \\    y & -x  \\ \end{matrix} \right]\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer92)
  The inverse of the matrix \[\left[ \begin{matrix}    5 & -2  \\    3 & 1  \\ \end{matrix} \right]\] is[Karnataka CET 2005]
  A)
  \[\frac{1}{11}\left[ \begin{matrix}    1 & 2  \\    -3 & 5  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    1 & 2  \\    -3 & 5  \\ \end{matrix} \right]\] done clear
  C)
  \[\frac{1}{13}\left[ \begin{matrix}    -2 & 5  \\    1 & 3  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & 3  \\    -2 & 5  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer93)
  \[A=\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 1  \\    0 & -2 & 4  \\ \end{matrix} \right];\,\,I=\left[ \begin{matrix}    1 & 0 & 0  \\    0 & 1 & 0  \\    0 & 0 & 1  \\ \end{matrix} \right]\]\[{{A}^{-1}}=\frac{1}{6}[{{A}^{2}}+cA+dI]\] where \[c,d\in R\], then pair of values \[(c,d)\] [IIT Screening 2005]
  A)
  (6, 11) done clear
  B)
  (6, -11) done clear
  C)
  (-6, 11) done clear
  D)
  (-6, -11) done clear
  View Solution play_arrow

• question_answer94)
  If \[P=\left[ \begin{matrix}    \frac{\sqrt{3}}{2} & \frac{1}{2}  \\    -\frac{1}{2} & \frac{\sqrt{3}}{2}  \\ \end{matrix} \right],\,A=\left[ \begin{matrix}    1 & 1  \\    0 & 1  \\ \end{matrix} \right]\] and \[Q=PA{{P}^{T}}\], then \[P({{Q}^{2005}}){{P}^{T}}\] equal to [IIT Screening 2005]
  A)
  \[\left[ \begin{matrix}    1 & 2005  \\    0 & 1  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    \sqrt{3}/2 & 2005  \\    1 & 0  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 2005  \\    \sqrt{3}/2 & 1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    1 & \sqrt{3}/2  \\    0 & 2005  \\ \end{matrix} \right]\] done clear
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• question_answer95)
  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 done clear
  B)
  25 done clear
  C)
  -1 done clear
  D)
  1 done clear
  E)
  125 done clear
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• question_answer96)
  If A is a unit matrix of order n, then \[A(adj\,A)\]is [DCE 2005]
  A)
  Zero matrix done clear
  B)
  Row matrix done clear
  C)
  Unit matrix done clear
  D)
  None of these done clear
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• question_answer97)
  If A is a skew-symmetric matrix of order n, and C is a column matrix of order \[n\times 1\], then \[{{C}^{T}}\]AC is [AMU 2005]
  A)
    A Identity matrix of order n done clear
  B)
  A unit matrix of order one done clear
  C)
  A zero matrix of order one done clear
  D)
  None of these done clear
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