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question_answer1)
Let A and B be two \[2\times 2\] matrices, Consider the statements |
(i) \[AB=O\Rightarrow A=O\] or \[B=0\] |
(ii) \[AB={{I}_{2}}\Rightarrow A={{B}^{-1}}\] |
(iii) \[{{(A+B)}^{2}}\]=\[{{A}^{2}}+2AB+{{B}^{2}}\] |
Then |
A)
(i) and (ii) are false, (iii) is true done
clear
B)
(ii) and (iii) are falsse, (i) is true done
clear
C)
(i) is false, (ii) and (iii) are true done
clear
D)
(i) and (iii) are false, (ii) is true done
clear
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question_answer2)
If A and B are two matrices such that \[AB=B\]and \[BA=A,\]then
A)
\[{{({{A}^{5}}-{{B}^{5}})}^{3}}=A-B\] done
clear
B)
\[{{({{A}^{5}}-{{B}^{5}})}^{3}}={{A}^{3}}-{{B}^{3}}\] done
clear
C)
\[A-B\]is idempotent done
clear
D)
\[A-B\]is nilpotent done
clear
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question_answer3)
If A is a square matrix such that \[{{A}^{2}}=A\], then \[{{(I+A)}^{3}}-7A\] is
A)
3I done
clear
B)
0 done
clear
C)
I done
clear
D)
2I done
clear
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question_answer4)
A is an involuntary matrix given by
then the inverse of A/2 will be
A)
2A done
clear
B)
\[\frac{{{A}^{-1}}}{2}\] done
clear
C)
\[\frac{A}{2}\] done
clear
D)
\[{{A}^{2}}\] done
clear
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question_answer5)
If both \[A-\frac{1}{2}I\] and \[A+\frac{1}{2}I\] are orthogonal matrices, then
A)
A is orthogonal done
clear
B)
A is skew-symmetric of even order done
clear
C)
\[{{A}^{2}}=\frac{3}{4}I\] done
clear
D)
none of these done
clear
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question_answer6)
If \[A=\left[ \begin{matrix} a & b \\ 0 & a \\ \end{matrix} \right]\] is nth root of \[{{I}_{2}}\], then choose the correct statements: |
(i) if n is odd, \[a=1,\text{ }b=0\] |
(ii) in n is odd, \[a=-1,\text{ }b=0\] |
(iii) if n is even, \[a=1,\text{ }b=0\] |
(iv) if n is even, \[a=-1,\text{ }b=0\] |
A)
i, ii, iii done
clear
B)
ii, iii, iv done
clear
C)
i, ii, iii, iv done
clear
D)
i, iii, iv done
clear
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question_answer7)
Let A be an nth-order square matrix and B be its adjoint, then \[\left| AB+K{{I}_{n}} \right|\]is (where K is a scalar quantity)
A)
\[{{(\left| A \right|+K)}^{n-2}}\] done
clear
B)
\[{{(\left| A \right|+K)}^{n}}\] done
clear
C)
\[{{(\left| A \right|+K)}^{n-1}}\] done
clear
D)
none of these done
clear
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question_answer8)
If
and if A is invertible, then which of the following is not true?
A)
\[\left| A \right|=\left| B \right|\] done
clear
B)
\[\left| A \right|=-\left| B \right|\] done
clear
C)
\[\left| adj\,A \right|=\left| adj\,B \right|\] done
clear
D)
A is invertible if and only if B is invertible done
clear
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question_answer9)
If A and B are two nonsingular matrices of the same order such that B?=I, for some positive integer r>1, then \[{{A}^{-1}}{{B}^{\,r-1}}A-{{A}^{-1}}{{B}^{-1}}A\]=
A)
I done
clear
B)
2I done
clear
C)
O done
clear
D)
-I done
clear
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question_answer10)
If \[A=\left[ \begin{matrix} 1 & \tan x \\ -\tan x & 1 \\ \end{matrix} \right]\], then \[{{A}^{T}}{{A}^{-1}}\]is
A)
\[\left[ \begin{matrix} -\cos 2x & \sin 2x \\ -\sin 2x & \cos 2x \\ \end{matrix} \right]\] done
clear
B)
\[\left[ \begin{matrix} \cos 2x & -\sin 2x \\ \sin 2x & \cos 2x \\ \end{matrix} \right]\] done
clear
C)
\[\left[ \begin{matrix} \cos 2x & \cos 2x \\ \sin 2x & \sin 2x \\ \end{matrix} \right]\] done
clear
D)
none of these done
clear
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question_answer11)
If
then \[A{{(\alpha ,\beta )}^{-1}}\]is equal to
A)
\[A(-\alpha ,-\beta )\] done
clear
B)
\[A(-\alpha ,\beta )\] done
clear
C)
\[A(\alpha ,-\beta )\] done
clear
D)
\[A(\alpha ,\beta )\] done
clear
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question_answer12)
If \[{{A}^{3}}=0\], then I=A+\[{{A}^{2}}\]equals
A)
\[I-A\] done
clear
B)
\[{{(I+{{A}^{1}})}^{-1}}\] done
clear
C)
\[{{(I-A)}^{-1}}\] done
clear
D)
none of these done
clear
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question_answer13)
If A and B are square matrices of the same order and A is nonsingular, then for a positive integer n, \[{{({{A}^{-1}}BA)}^{n}}\] is equal
A)
\[{{A}^{-n}}{{B}^{n}}{{A}^{n}}\] done
clear
B)
\[{{A}^{n}}{{B}^{n}}{{A}^{-n}}\] done
clear
C)
\[{{A}^{-1}}{{B}^{n}}A\] done
clear
D)
\[n({{A}^{-1}}BA)\] done
clear
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question_answer14)
\[{{(-A)}^{-1}}\] is always equal to (where A is nth-order square matrix)
A)
\[{{(-1)}^{n}}{{A}^{-1}}\] done
clear
B)
\[-{{A}^{-1}}\] done
clear
C)
\[{{(-1)}^{n-1}}{{A}^{-1}}\] done
clear
D)
none of these done
clear
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question_answer15)
If \[\left[ \begin{matrix} 2 & 1 \\ 3 & 2 \\ \end{matrix} \right]A\left[ \begin{matrix} -3 & 2 \\ 5 & -3 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]\], then A=
A)
\[\left[ \begin{matrix} 1 & 1 \\ 1 & 0 \\ \end{matrix} \right]\] done
clear
B)
\[\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right]\] done
clear
C)
\[\left[ \begin{matrix} 1 & 0 \\ 1 & 1 \\ \end{matrix} \right]\] done
clear
D)
\[-\left[ \begin{matrix} 1 & 1 \\ 1 & 0 \\ \end{matrix} \right]\] done
clear
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question_answer16)
If A is symmetric as well as skew-symmetric matrix, then A is
A)
diagonal matrix done
clear
B)
null matrix done
clear
C)
triangular matrix done
clear
D)
none of these done
clear
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question_answer17)
The number of solutions of the matrix equation\[{{X}^{2}}=\left[ \begin{matrix} 1 & 1 \\ 2 & 3 \\ \end{matrix} \right]\] is
A)
more than 2 done
clear
B)
2 done
clear
C)
0 done
clear
D)
1 done
clear
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question_answer18)
Given that matrix
. If \[xyz=60\] and \[8x+4y+3z=20,\] then A (adjA) is equal to
A)
B)
C)
D)
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question_answer19)
Elements of a matrix A of order \[10\times 10\] are defined is \[{{a}_{ij}}={{w}^{i+j}}\] (where w is cube root of unity), then tr of the matrix is
A)
0 done
clear
B)
1 done
clear
C)
3 done
clear
D)
none of these done
clear
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question_answer20)
If \[A=\left[ \begin{matrix} a & b \\ b & a \\ \end{matrix} \right]\]and \[{{A}^{2}}=\left[ \begin{matrix} \alpha & \beta \\ \beta & \alpha \\ \end{matrix} \right]\], then
A)
\[\alpha ={{a}^{2}}+{{b}^{2}},\beta =ab\] done
clear
B)
\[\alpha ={{a}^{2}}+{{b}^{2}},\beta =2ab\] done
clear
C)
\[\alpha ={{a}^{2}}+{{b}^{2}},\beta ={{a}^{2}}-{{b}^{2}}\] done
clear
D)
\[\alpha =2ab,\,\,\beta ={{a}^{2}}+{{b}^{2}}\] done
clear
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question_answer21)
Let \[A=\left[ \begin{matrix} 0 & \alpha \\ 0 & 0 \\ \end{matrix} \right]\] and \[{{(A+1)}^{50}}-50A=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]\].Then the value of a+b+c+d is _____.
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question_answer22)
If matrix A is given by \[A=\left[ \begin{matrix} 6 & 11 \\ 2 & 4 \\ \end{matrix} \right]\], then the determinant of \[{{A}^{2005}}-6{{A}^{2004}}\] is _______.
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question_answer23)
If \[A={{[{{a}_{i\,j}}]}_{4\,\times \,4}},\] such that \[{{a}_{ij}}=\left\{ \begin{matrix} 2,\,\,when\,\,\,i=j \\ 0,\,\,when\,\,\,i\ne j \\ \end{matrix} \right.\] then \[\left\{ \frac{\det (adj(adj\,A))}{7} \right\}\] is (where {.} represents fractional part function) ______.
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question_answer24)
If A is nonsingular and \[(A-2I)\]\[(A-4I)=O\], then \[\frac{1}{6}A+\frac{4}{3}{{A}^{-1}}\] is equal to KI. Then the value of k is _______.
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question_answer25)
Let
and
. If B is the inverse of A, then \[\alpha \] is _____.
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