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Recent questions and answers in Linear Algebra
40
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14
answers
1
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
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Linear Algebra
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ago
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artrides
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23
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5
answers
2
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 ____
Rohit139
answered
in
Linear Algebra
Mar 10
by
Rohit139
10.0k
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gatecse-2018
linear-algebra
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normal
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17
votes
4
answers
3
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
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in
Linear Algebra
Mar 10
by
Rohit139
9.7k
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gatecse-2022
linear-algebra
matrix
1-mark
29
votes
8
answers
4
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
17.6k
views
gateit-2005
linear-algebra
normal
determinant
53
votes
7
answers
5
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
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in
Linear Algebra
Feb 25
by
SASIDHAR_1
15.5k
views
gatecse-2016-set2
linear-algebra
system-of-equations
normal
19
votes
4
answers
6
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
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in
Linear Algebra
Feb 25
by
SASIDHAR_1
6.2k
views
gatecse-2005
linear-algebra
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normal
48
votes
7
answers
7
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
21.2k
views
gate1996
linear-algebra
system-of-equations
normal
0
votes
1
answer
8
#GATE
Is Vector Subspace, Span, Basis, Dimension part of the gate CSE engineering mathematics linear algebra syllabus ?
NandanKumar07
answered
in
Linear Algebra
Feb 20
by
NandanKumar07
112
views
0
votes
1
answer
9
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 20
by
gate.datascience_ai
153
views
gate2024-da-memory-based
goclasses
linear-algebra
vector-space
numerical-answers
3
votes
7
answers
10
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
2.5k
views
gatecse2024-set1
linear-algebra
6
votes
1
answer
11
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
2.4k
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multiple-selects
linear-algebra
1
vote
0
answers
12
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
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in
Linear Algebra
Feb 16
by
Arjun
1.7k
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gatecse2024-set2
linear-algebra
multiple-selects
5
votes
2
answers
13
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
472
views
goclasses2024-mockgate-14
linear-algebra
system-of-equations
multiple-selects
2-marks
2
votes
2
answers
14
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
385
views
gate2024-da-memory-based
goclasses
linear-algebra
determinant
numerical-answers
0
votes
2
answers
15
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
216
views
gate2024-da-memory-based
goclasses
linear-algebra
eigen-value
5
votes
1
answer
16
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
303
views
goclasses2024-mockgate-14
linear-algebra
rank-of-matrix
multiple-selects
1-mark
0
votes
0
answers
17
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 5
by
GO Classes
190
views
gate2024-da-memory-based
goclasses
linear-algebra
vector-space
0
votes
0
answers
18
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?
GO Classes
asked
in
Linear Algebra
Feb 5
by
GO Classes
89
views
gate2024-da-memory-based
goclasses
linear-algebra
system-of-equations
0
votes
0
answers
19
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 5
by
GO Classes
95
views
gate2024-da-memory-based
goclasses
linear-algebra
vector-space
9
votes
2
answers
20
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
447
views
goclasses2024-mockgate-13
goclasses
linear-algebra
eigen-vector
2-marks
7
votes
1
answer
21
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
525
views
goclasses2024-mockgate-13
goclasses
linear-algebra
matrix
1-mark
22
votes
4
answers
22
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
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in
Linear Algebra
Jan 28
by
me.himanshu.k
8.4k
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gatecse-2010
linear-algebra
eigen-value
easy
0
votes
3
answers
23
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
1.3k
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gateda-sample-paper-2024
linear-algebra
eigen-value
7
votes
2
answers
24
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
560
views
goclasses2024-mockgate-12
goclasses
linear-algebra
rank-of-matrix
multiple-selects
1-mark
0
votes
2
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25
GATE 2022 | Statistics | Question 47
Let 𝑴 be any 3 × 3 symmetric matrix with eigenvalues 1, 2 and 3. Let 𝑵 be any 3 × 3 matrix with real eigenvalues such that $𝑴𝑵 + 𝑵^T𝑴 = 3𝑰$, where 𝑰 is the 3 × 3 identity matrix. Then which of the following cannot be eigenvalue(s) of the matrix 𝑵 ? (A) 1/4 (B) 3/4 (C) 1/2 (D) 7/4
Krutarth.8
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in
Linear Algebra
Jan 15
by
Krutarth.8
143
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linear-algebra
5
votes
1
answer
26
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 $m \times 1, m \times n$ ... $\operatorname{rank}[A]>$ $\operatorname{rank}[A \mid b]$.
GauravRajpurohit
answered
in
Linear Algebra
Jan 14
by
GauravRajpurohit
446
views
goclasses2024-mockgate-11
goclasses
linear-algebra
system-of-equations
multiple-selects
2-marks
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
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Linear Algebra
Jan 13
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GO Classes
678
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goclasses2024-mockgate-11
goclasses
numerical-answers
linear-algebra
eigen-value
1-mark
0
votes
0
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28
Linear Transformation of Matrix
Debargha Mitra Roy
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Linear Algebra
Jan 12
by
Debargha Mitra Roy
56
views
linear-algebra
matrix
1
vote
1
answer
29
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
sunnyb142
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in
Linear Algebra
Jan 12
by
sunnyb142
123
views
linear-algebra
vector-space
0
votes
1
answer
30
GATE 2022 | MATHS | Q-65
Let M be a $3 × 3$ real matrix such that $M^2 = 2M + 3I$. If the determinant of $M$ is $−9$, then the trace of $M$ equals._______
ssingla
answered
in
Linear Algebra
Jan 11
by
ssingla
119
views
linear-algebra
0
votes
1
answer
31
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 ________________________
ssingla
answered
in
Linear Algebra
Jan 11
by
ssingla
137
views
linear-algebra
0
votes
2
answers
32
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 \[ | \lambda - \mu | = \] _________
ssingla
answered
in
Linear Algebra
Jan 11
by
ssingla
84
views
linear-algebra
0
votes
1
answer
33
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
ssingla
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in
Linear Algebra
Jan 11
by
ssingla
136
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vector-space
0
votes
0
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34
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
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in
Linear Algebra
Jan 11
by
rajveer43
141
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linear-algebra
0
votes
0
answers
35
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
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in
Linear Algebra
Jan 11
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rajveer43
80
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linear-algebra
0
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36
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
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in
Linear Algebra
Jan 11
by
rajveer43
64
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linear-algebra
0
votes
0
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37
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
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in
Linear Algebra
Jan 11
by
rajveer43
78
views
linear-algebra
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
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in
Linear Algebra
Jan 11
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rajveer43
54
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linear-algebra
vector-space
0
votes
1
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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
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Jan 11
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rajveer43
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linear-algebra
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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
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