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Recent questions and answers in Linear Algebra
23
votes
6
answers
1
GATE CSE 2018 | Question: 17
Consider a matrix $A= uv^T$ where $u=\begin{pmatrix}1 \\ 2 \end{pmatrix} , v = \begin{pmatrix}1 \\1 \end{pmatrix}$. Note that $v^T$ denotes the transpose of $v$. The largest eigenvalue of $A$ is ____
dharmik_3103
answered
in
Linear Algebra
Mar 23
by
dharmik_3103
10.1k
views
gatecse-2018
linear-algebra
eigen-value
normal
numerical-answers
1-mark
3
votes
0
answers
2
#linear algebra
why Eigen Vectors can not be zero ?
srijankarak_123
asked
in
Linear Algebra
Mar 23
by
srijankarak_123
25
views
49
votes
10
answers
3
GATE CSE 2016 Set 1 | Question: 05
Two eigenvalues of a $3 \times 3$ real matrix $P$ are $(2+\sqrt {-1})$ and $3$. The determinant of $P$ is _______
reccurGautam
answered
in
Linear Algebra
Mar 20
by
reccurGautam
14.5k
views
gatecse-2016-set1
linear-algebra
eigen-value
numerical-answers
normal
0
votes
1
answer
4
Memory Based GATE DA 2024 | Question: 53
Consider the following scenarios involving linear algebra: For a \(3 \times 3\) matrix, if some vector p has a unique solution, can there exist another vector q with an infinite solution? For a \(3 \times 3\) matrix, if some vector p ... 2 \times 3\) matrix, if some vector p has a unique solution, can there exist another vector q with an infinite solution?
Ajay Sreenivas
answered
in
Linear Algebra
Mar 19
by
Ajay Sreenivas
117
views
gate2024-da-memory-based
goclasses
linear-algebra
system-of-equations
43
votes
14
answers
5
GATE CSE 2021 Set 2 | Question: 24
Suppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $\text{Px=0}$ is a scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T$. The rank of $P$ is __________
artrides
answered
in
Linear Algebra
Mar 18
by
artrides
18.1k
views
gatecse-2021-set2
numerical-answers
linear-algebra
matrix
rank-of-matrix
1-mark
18
votes
4
answers
6
GATE CSE 2022 | Question: 10
Consider the following two statements with respect to the matrices $\textit{A}_{m \times n}, \textit{B}_{n \times m}, \textit{C}_{n \times n}$ and $ \textit{D}_{n \times n}.$ Statement $1: tr \text{(AB)} = tr \text{(BA)}$ ... $2$ is correct. Both Statement $1$ and Statement $2$ are correct. Both Statement $1$ and Statement $2$ are wrong.
Rohit139
answered
in
Linear Algebra
Mar 10
by
Rohit139
10.4k
views
gatecse-2022
linear-algebra
matrix
1-mark
29
votes
8
answers
7
GATE IT 2005 | Question: 3
The determinant of the matrix given below is $\begin{bmatrix} 0 &1 &0 &2 \\ -1& 1& 1& 3\\ 0&0 &0 & 1\\ 1& -2& 0& 1 \end{bmatrix}$ $-1$ $0$ $1$ $2$
EagerLearner
answered
in
Linear Algebra
Mar 8
by
EagerLearner
20.3k
views
gateit-2005
linear-algebra
normal
determinant
54
votes
7
answers
8
GATE CSE 2016 Set 2 | Question: 04
Consider the systems, each consisting of $m$ linear equations in $n$ variables. If $m < n$, then all such systems have a solution. If $m > n$, then none of these systems has a solution. If $m = n$, then there exists a system which has a solution. ... $\text{II}$ and $\text{III}$ are true. Only $\text{III}$ is true. None of them is true.
SASIDHAR_1
answered
in
Linear Algebra
Feb 25
by
SASIDHAR_1
15.7k
views
gatecse-2016-set2
linear-algebra
system-of-equations
normal
19
votes
4
answers
9
GATE CSE 2005 | Question: 48
Consider the following system of linear equations : $2x_1 - x_2 + 3x_3 = 1$ $3x_1 + 2x_2 + 5x_3 = 2$ $-x_1+4x_2+x_3 = 3$ The system of equations has no solution a unique solution more than one but a finite number of solutions an infinite number of solutions
SASIDHAR_1
answered
in
Linear Algebra
Feb 25
by
SASIDHAR_1
7.2k
views
gatecse-2005
linear-algebra
system-of-equations
normal
48
votes
7
answers
10
GATE CSE 1996 | Question: 1.7
Let $Ax = b$ be a system of linear equations where $A$ is an $m \times n$ matrix and $b$ is a $m \times 1$ column vector and $X$ is an $n \times1$ column vector of unknowns. Which of the following is false? The system has a solution if and ... a unique solution. The system will have only a trivial solution when $m=n$, $b$ is the zero vector and $\text{rank}(A) =n$.
SASIDHAR_1
answered
in
Linear Algebra
Feb 25
by
SASIDHAR_1
21.3k
views
gate1996
linear-algebra
system-of-equations
normal
0
votes
1
answer
11
#GATE
Is Vector Subspace, Span, Basis, Dimension part of the gate CSE engineering mathematics linear algebra syllabus ?
NandanKumar07
answered
in
Linear Algebra
Feb 19
by
NandanKumar07
132
views
0
votes
1
answer
12
Memory Based GATE DA 2024 | Question: 29
Consider the vector \( u = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{bmatrix} \), and let \( M = uu^{\top} \). If \( \sigma_1, \sigma_2, \sigma_3, \ldots, \sigma_5 \) are the singular values of \( M \), what is the value of \( \sum_{i=1}^5 \sigma_i \)?
gate.datascience_ai
answered
in
Linear Algebra
Feb 19
by
gate.datascience_ai
170
views
gate2024-da-memory-based
goclasses
linear-algebra
vector-space
numerical-answers
3
votes
7
answers
13
GATE CSE 2024 | Set 1 | Question: 2
The product of all eigenvalues of the matrix $\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$ is $-1$ $0$ $1$ $2$
TusharRana
answered
in
Linear Algebra
Feb 17
by
TusharRana
2.6k
views
gatecse2024-set1
linear-algebra
6
votes
1
answer
14
GATE CSE 2024 | Set 1 | Question: 39
Let $A$ be any $n \times m$ matrix, where $m>n$. Which of the following statements is/are TRUE about the system of linear equations $Ax=0$? There exist at least $m-n$ linearly independent solutions to this system There exist $m-n$ ... solution in which at least $m-n$ variables are $0$ There exists a solution in which at least $n$ variables are non-zero
Sachin Mittal 1
answered
in
Linear Algebra
Feb 17
by
Sachin Mittal 1
2.6k
views
gatecse2024-set1
multiple-selects
linear-algebra
2
votes
0
answers
15
GATE CSE 2024 | Set 2 | Question: 37
Let $A$ be an $n \times n$ matrix over the set of all real numbers $\mathbb{R}$. Let $B$ be a matrix obtained from $A$ by swapping two rows. Which of the following statements is/are TRUE? The determinant of $B$ is the negative of the ... If $A$ is symmetric, then $B$ is also symmetric If the trace of $A$ is zero, then the trace of $B$ is also zero
Arjun
asked
in
Linear Algebra
Feb 16
by
Arjun
1.8k
views
gatecse2024-set2
linear-algebra
multiple-selects
6
votes
2
answers
16
GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 49
Consider a $2 \times 2$ matrix M. Which of the following are NOT POSSIBLE for the system of equations $M x=p?$ no solutions for some but not all $\vec{p}$; exactly one solution for all other $\vec{p}$ exactly one solution for ... some $\vec{p}$, exactly one solution for some $\vec{p}$ and more than one solution for some $\vec{p}$
Extra_Sauce
answered
in
Linear Algebra
Feb 7
by
Extra_Sauce
513
views
goclasses2024-mockgate-14
linear-algebra
system-of-equations
multiple-selects
2-marks
2
votes
2
answers
17
Memory Based GATE DA 2024 | Question: 4
Consider the matrix \[ \mathrm{M} = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6 \end{bmatrix} \] Find the value of \(|\mathrm{M}^2 + 12M|\).
krishnajsw
answered
in
Linear Algebra
Feb 7
by
krishnajsw
444
views
gate2024-da-memory-based
goclasses
linear-algebra
determinant
numerical-answers
0
votes
2
answers
18
Memory Based GATE DA 2024 | Question: 6
Consider the matrix \[ \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix} \] What is the nature of the eigenvalues of the given matrix? Both eigenvalues are positive. One eigenvalue is negative. Eigenvalues are complex conjugate pairs. None of the above.
pankaj kumar 70m
answered
in
Linear Algebra
Feb 7
by
pankaj kumar 70m
242
views
gate2024-da-memory-based
goclasses
linear-algebra
eigen-value
5
votes
1
answer
19
GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 22
Let $A$ be a $20 \times 11$ matrix with real entries. After performing some row operations on $A$, we get a matrix $B$ which has 12 nonzero rows. Which of the following is/are always true? The rank of $A$ is 12. The ranks of $A$ and $B$ are ... . If $v$ is a vector such that $A v=0$ then $B v$ is also 0. The rank of $B$ is at most 11.
GO Classes
answered
in
Linear Algebra
Feb 5
by
GO Classes
322
views
goclasses2024-mockgate-14
linear-algebra
rank-of-matrix
multiple-selects
1-mark
0
votes
0
answers
20
Memory Based GATE DA 2024 | Question: 5
Consider a matrix \(M \in \mathbb{R}^{3 \times 3}\) and let \(U\) be a 2-dimensional subspace such that \(M\) is a projection onto \(U\). Which of the following statements are true? \(M^3 = M\) \(M^2 = M\) The nullspace of \(M\) is 1-dimensional. The nullspace of \(M\) is 2-dimensional.
GO Classes
asked
in
Linear Algebra
Feb 4
by
GO Classes
201
views
gate2024-da-memory-based
goclasses
linear-algebra
vector-space
0
votes
0
answers
21
Memory Based GATE DA 2024 | Question: 60
Linear Algebra Question: Four options were given related to subspace R3. Something like this : A. \( \alpha \cdot x + \beta \cdot y \) B. \( \alpha^2 \cdot x + \beta^2 \cdot y \) C. \(f(x) = 4x_1 + 2x_3 + 3x_3 \) D.
GO Classes
asked
in
Linear Algebra
Feb 4
by
GO Classes
109
views
gate2024-da-memory-based
goclasses
linear-algebra
vector-space
9
votes
2
answers
22
GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 43
Let $A$ be a $2 \times 2$ matrix for which there is a constant $k$ such that the sum of the entries in each row and each column is $k$. Which of the following must be an eigenvector of $A?$ ... $\left[\begin{array}{l}1 \\ 1\end{array}\right]$. I only II only III only I and II only
Saiteja529
answered
in
Linear Algebra
Jan 30
by
Saiteja529
468
views
goclasses2024-mockgate-13
goclasses
linear-algebra
eigen-vector
2-marks
7
votes
1
answer
23
GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 13
If $A$ is a $3 \times 3$ matrix such that $A\left(\begin{array}{l}0 \\ 1 \\ 2\end{array}\right)=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$ ... $\left(\begin{array}{r}1 \\ -1 \\ 0\end{array}\right)$ $\left(\begin{array}{r}9 \\ 10 \\ 11\end{array}\right)$
SankarVinayak
answered
in
Linear Algebra
Jan 29
by
SankarVinayak
551
views
goclasses2024-mockgate-13
goclasses
linear-algebra
matrix
1-mark
22
votes
4
answers
24
GATE CSE 2010 | Question: 29
Consider the following matrix $A = \left[\begin{array}{cc}2 & 3\\x & y \end{array}\right]$ If the eigenvalues of A are $4$ and $8$, then $x = 4$, $y = 10$ $x = 5$, $y = 8$ $x = 3$, $y = 9$ $x = -4$, $y =10$
me.himanshu.k
answered
in
Linear Algebra
Jan 27
by
me.himanshu.k
8.5k
views
gatecse-2010
linear-algebra
eigen-value
easy
0
votes
3
answers
25
GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 17
For matrix $H=\left[\begin{array}{cc}9 & -2 \\ -2 & 6\end{array}\right]$, one of the eigenvalues is $5$. Then, the other eigenvalue is $12$ $10$ $8$ $6$
Arjunmaniya
answered
in
Linear Algebra
Jan 25
by
Arjunmaniya
1.4k
views
gateda-sample-paper-2024
linear-algebra
eigen-value
7
votes
2
answers
26
GO Classes Test Series 2024 | Mock GATE | Test 12 | Question: 7
Let $\text{A}$ be a $20 \times 11$ matrix with real entries. After performing some row operations on $\text{A}$, we get a matrix $\text{B}$ which has $12$ nonzero rows. Which of the following is/are always true? The rank of $\text{A}$ ... that $\text{A} v=0$ then $\text{B} v$ is also $0.$ The rank of $\text{B}$ is at most $11.$
GauravRajpurohit
answered
in
Linear Algebra
Jan 21
by
GauravRajpurohit
572
views
goclasses2024-mockgate-12
goclasses
linear-algebra
rank-of-matrix
multiple-selects
1-mark
4
votes
0
answers
27
GO Classes Test Series 2024 | Mock GATE | Test 11 | Question: 30
Let $A$ be a matrix defined as $A=u v^T$, where $u$ and $v$ are column vectors of dimension $3 \times 1$. The resulting matrix $A$ will be of dimension $3 \times 3$. What are the maximum number of nonzero eigenvalues possible for the matrix $A?$
GO Classes
asked
in
Linear Algebra
Jan 13
by
GO Classes
702
views
goclasses2024-mockgate-11
goclasses
numerical-answers
linear-algebra
eigen-value
1-mark
5
votes
1
answer
28
GO Classes Test Series 2024 | Mock GATE | Test 11 | Question: 36
Consider the system $A \mathbf{x}=\mathbf{b}$, with coefficient matrix $A$ and augmented matrix $[A \mid b]$. The sizes of $\mathbf{b}, A$, and $[A \mid \mathbf{b}]$ are $m \times 1, m \times n$ ... $\operatorname{rank}[A]>$ $\operatorname{rank}[A \mid b]$.
GO Classes
asked
in
Linear Algebra
Jan 13
by
GO Classes
479
views
goclasses2024-mockgate-11
goclasses
linear-algebra
system-of-equations
multiple-selects
2-marks
0
votes
0
answers
29
Linear Transformation of Matrix
Debargha Mitra Roy
asked
in
Linear Algebra
Jan 12
by
Debargha Mitra Roy
62
views
linear-algebra
matrix
0
votes
1
answer
30
GATE 2016 | MATHS | Q-48
Let \( M= \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \) and \( e^M = Id + M + \frac{M^2}{2!} + \frac{M^3}{3!} + \frac{M^4}{4!} + \ldots \). If \( e^M = [b_{ij}]\). then \[\frac{1}{e} \sum_{i=1}^{3} \sum_{j=1}^{3} b_{ij} \] is equal to ________________________
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
150
views
linear-algebra
0
votes
0
answers
31
GATE 2016 | MATHS | QUESTION-47
Let \( A = \begin{bmatrix} a & b & c \\ b& d & e\\ c& e& f\end{bmatrix} \) be a real matrix with eigenvalues 1, 0, and 3. If the eigenvectors corresponding to 1 and 0 are \(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\) and \(\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}\) respectively, then the value of \(3f\) is equal to ________________________
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
156
views
linear-algebra
0
votes
1
answer
32
GATE 2016 | MATHS | Q-11
Let \( \mathbf{v}, \mathbf{w}, \mathbf{u} \) be a basis of \( \mathbb{V} \). Consider the following statements P and Q: (P) : \( \{\mathbf{v} + \mathbf{w}, \mathbf{w} + \mathbf{u}, \mathbf{v} - \mathbf{u}\} \) is a basis of \( \mathbb{V} \). ( ... a basis of \( \mathbb{V} \). Which of the above statements hold TRUE? (A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
145
views
vector-space
0
votes
0
answers
33
GATE 2016 | MATHS | Q-12
Consider the following statements P and Q: (P) : If \( M = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{bmatrix} \), then M is singular. (Q) : Let S be a diagonalizable matrix. If T is a matrix such that \( ... ), then T is diagonalizable. Which of the above statements hold TRUE? (A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
87
views
linear-algebra
0
votes
0
answers
34
GATE 2016 | MATHS | Q-14
Consider a real vector space \( V \) of dimension \( n \) and a non-zero linear transformation \( T: \mathbb{V} \rightarrow \mathbb{V} \). If \( \text{dim}(T) < n \) and $T^2 = \lambda T$, for some \( \lambda \in \mathbb{R} \backslash \{0\} \), then which ... 0 \) for all \( X \in \mathbb{W} \) (C) \( T \) is invertible (D) \( \lambda\) is the only eigenvalue of \( T \)
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
71
views
linear-algebra
0
votes
0
answers
35
GATE 2018 | MATHS | Q-55
Let \( A \) be a \(3 \times 3\) matrix with real entries. If three solutions of the linear system of differential equations \(\dot{x}(t) = Ax(t)\) are given by \[ \begin{bmatrix} e^t - e^{2t} \\ -e^{t} + e^{2t} \\ e^t + e^{2t} \end{bmatrix}, \begin{bmatrix} ... \\ e^{-t} - 2e^t \\ -e^{-t} + 2e^t \end{bmatrix}, \] then the sum of the diagonal entries of \( A \) is equal to
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
86
views
linear-algebra
0
votes
2
answers
36
GATE 2018 | MATHS | Q-52
Consider the matrix \( A = I_9 - 2u^T u \) with \( u = \frac{1}{3}[1, 1, 1, 1, 1, 1, 1, 1, 1] \), where \( I_9 \) is the \(9 \times 9\) identity matrix and \( u^T \) is the transpose of \( u \). If \( \lambda \) and \( \mu \) are two distinct eigenvalues of \( A \), then \[ | \lambda - \mu | = \] _________
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
97
views
linear-algebra
1
vote
1
answer
37
GATE 2018 | MATHS | Q-51
Consider \( \mathbb{R}^3 \) with the usual inner product. If \( d \) is the distance from \( (1, 1, 1) \) to the subspace ${(1, 1, 0), (0, 1, 1)}$ of \( \mathbb{R}^3 \), then \( 3d^2 \) is given by
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
132
views
linear-algebra
vector-space
0
votes
0
answers
38
GATE 2018 | MATHS | Q-50
Let \( M_2(\mathbb{R}) \) be the vector space of all \( 2 \times 2 \) real matrices over the field \( \mathbb{R} \). Define the linear transformation \( S : M_2(\mathbb{R}) \to M_2(\mathbb{R}) \) by \( S(X) = 2X + X^T \), where \( X^T \) denotes the transpose of the matrix \( X \). Then the trace of \( S \) equals________
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
61
views
linear-algebra
vector-space
0
votes
1
answer
39
GATE 2018 | MATHS | Q-42 DA Practice Questions
Consider the following two statements: \(P\): The matrix \(\begin{bmatrix} 0 & 5 \\ 0 & 7 \end{bmatrix}\) has infinitely many LU factorizations, where \(L\) is lower triangular with each diagonal entry 1 and \(U\) is upper triangular. \(Q\): The matrix \( ... \(Q\) are TRUE (C) \(P\) is FALSE and \(Q\) is TRUE (D) Both \(P\) and \(Q\) are FALSE
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
166
views
linear-algebra
0
votes
0
answers
40
GATE 2018 | MATHS | Q-24
Consider the subspaces \[ W_1 = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1 = x_2 + 2x_3 \} \] \[ W_2 = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1 = 3x_2 + 2x_3 \} \] of \( \mathbb{R}^3 \). Then the dimension of \(W_1 + W_2\) equals_________
rajveer43
asked
in
Linear Algebra
Jan 11
by
rajveer43
49
views
linear-algebra
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