Recent questions tagged matrix / GATE Overflow for GATE CSE

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Linear Algebra AX=B
Consider a matrix A (n×m) ,X(m×n) and B(n×n) such that AX=B . If A has k linearly independent columns then what conclusions can we nake about the number of linearly independent columns of B.

Consider a matrix A (n×m) ,X(m×n) and B(n×n) such that AX=B . If A has k linearly independent columns then what conclusions can we nake about the number of linearly in...

Soumya04

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Soumya04 asked 3 days ago

6 votes

1 answer

2

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 1
Let $A$ be an $n \times n$ matrix of real or complex numbers. Which of the following statements are equivalent to: “the matrix $A$ is invertible”? The columns of $A$ are linearly independent. The rows of $A$ are linearly independent. The only solution of the homogeneous equations $Ax = 0$ is $x = 0$. The rank of $A$ is $n$.

Let $A$ be an $n \times n$ matrix of real or complex numbers. Which of the following statements are equivalent to: “the matrix $A$ is invertible”?The columns of $A$ a...

GO Classes

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GO Classes asked Apr 3

7 votes

3 answers

3

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 2
Let $M$ be a $2 \times 2$ matrix with the property that the sum of each of the rows and also the sum of each of the columns is the same constant $c$. Which (if any) any of the vectors must be an eigenvector of $M$ ... $W = \left[\begin{array}{l}1 \\ 1 \\ \end{array}\right]$ $U$ $V$ $W$ None of the above

Let $M$ be a $2 \times 2$ matrix with the property that the sum of each of the rowsand also the sum of each of the columns is the same constant $c$. Which (if any) any of...

GO Classes

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GO Classes asked Apr 3

3 votes

1 answer

4

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 3
Let the $n \times n$ matrix $A$ have an eigenvalue $\lambda$ with corresponding eigenvector $v$. Which of the following statements are true for matrix $A$. $-v$ is an eigenvector of $-A$ with eigenvalue $- \lambda$. If $v$ is also ... $A+B$. eigenvalue of $A^3$ is $\lambda^3$ and the eigenvector is $v^3$ .

Let the $n \times n$ matrix $A$ have an eigenvalue $\lambda$ with corresponding eigenvector $v$.Which of the following statements are true for matrix $A$.$-v$ is an eigen...

GO Classes

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GO Classes asked Apr 3

1 votes

2 answers

5

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 4
Consider the following matrix A: $\left[\begin{array}{lll}2 & -1 & 0 \\ 0 & 2 & 0 \\ 1 & 0 & 2\end{array}\right]$ Which of the following regarding the matrix $A$ ... $x$ is the eigenvector corresponding to eigenvalue $\lambda$ of $A$ then $x$ is also the eigenvector of $A^{-1}$.

Consider the following matrix A:$\left[\begin{array}{lll}2 & -1 & 0 \\ 0 & 2 & 0 \\ 1 & 0 & 2\end{array}\right]$Which of the following regarding the matrix $A$ is/are cor...

GO Classes

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GO Classes asked Apr 3

1 votes

1 answer

6

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 5
Suppose a $3 \times 5$ matrix $A$ has rank $r = 3$. Then the equation $Ax = b$ $\textbf{BLANK 1}$ has $\textbf{BLANK 2}$ ... BLANK 2: Infinitely many solutions BLANK 1: Sometimes, BLANK 2: Unique solution BLANK 1: Sometimes, BLANK 2: Infinitely many solutions

Suppose a $3 \times 5$ matrix $A$ has rank $r = 3$. Then the equation $Ax = b$ $\textbf{BLANK 1}$ has $\textbf{BLANK 2}$.Which of the following are appropriate words ...

GO Classes

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GO Classes asked Apr 3

3 votes

2 answers

7

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 6
Which of the following statements is/are $\textbf{NOT CORRECT}$? If $v1$ and $v2$ are linearly independent eigenvectors then they can correspond to the same eigenvalue. If $A$ is a nilpotent matrix, meaning that $A^k = 0$ for the ... is an eigenvalue of an invertible matrix $A$ then $\lambda ^{-1}$ is an eigenvalue of $A^{-1}$.

Which of the following statements is/are $\textbf{NOT CORRECT}$?If $v1$ and $v2$ are linearly independent eigenvectors then they can correspond to the same eigenvalue.If ...

GO Classes

71 views

GO Classes asked Apr 3

2 votes

1 answer

8

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 7
Given an $m \times n$ matrix $A$ whose rows are linearly independent. Now, consider following statements regarding $A$: $S1:$ The system of equations $Ax = b$ for any $b$ is consistent. $S2:$ $Ax = b$ always has a unique solution. ... $S2$ is FALSE. $S1$ is FALSE and $S2$ is TRUE. Both $S1$ and $S2$ are FALSE.

Given an $m \times n$ matrix $A$ whose rows are linearly independent. Now, consider following statements regarding $A$: $S1:$ The system of equations $Ax = b$ for any ...

GO Classes

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GO Classes asked Apr 3

3 votes

1 answer

9

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 8
Which of the following statements is/are $\textbf{FALSE}$? For $n \times n$ real-symmetric matrices $A$ and $B$, $AB$ and $BA$ always have the same eigenvalues. For $n \times n$ matrices $A$ and $B$ with $B$ ... eigenvectors. For $n \times n$ real-symmetric matrices $A$ and $B$, $AB$ and $BA$ always have the same eigenvectors.

Which of the following statements is/are $\textbf{FALSE}$?For $n \times n$ real-symmetric matrices $A$ and $B$, $AB$ and $BA$ always have the same eigenvalues.For $n \tim...

GO Classes

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GO Classes asked Apr 3

4 votes

1 answer

10

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 9
Let $A$ and $B$ be two $n \times n$ matrices. If $B$ is invertible and $(I+BA)^{-1} = 2B^2$, then which of the following is the correct definition of $A$ in terms of $B$? $A = (1/2)B^{-3} - B^{-1}$ $A = 2B^{-3}- B^{-1}$ $A = 2B^3-I$ None of the above

Let $A$ and $B$ be two $n \times n$ matrices. If $B$ is invertible and $(I+BA)^{-1} = 2B^2$, then which of the following is the correct definition of $A$ in terms of $B$?...

GO Classes

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GO Classes asked Apr 3

3 votes

1 answer

11

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 10
A $4 \times 4$ matrix $\mathrm{A}$ has rank 3 . Which of the following is/are true? 1. $A^{-1}$ does not exist 2. $A^{-1}$ may exist, and if it does, its rank must be less than 3 3. $A^{-1}$ may ... , and it can take any rank less than 5 Only 1 is correct Only 2,3 are correct Only 4 is correct None of the statements are correct.

A $4 \times 4$ matrix $\mathrm{A}$ has rank 3 . Which of the following is/are true?1. $A^{-1}$ does not exist2. $A^{-1}$ may exist, and if it does, its rank must be less ...

GO Classes

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GO Classes asked Apr 3

1 votes

1 answer

12

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 11
The rank and nullity of a matrix $A$ are 4 and 2 , respectively. The nullity of $A^{\top}$ is 3 . What are the dimensions of $A$ ? $6 \times 7$ $4 \times 5$ $7 \times 6$ $5 \times 4$

The rank and nullity of a matrix $A$ are 4 and 2 , respectively. The nullity of $A^{\top}$ is 3 . What are the dimensions of $A$ ?$6 \times 7$$4 \times 5$$7 \times 6$$5 \...

GO Classes

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GO Classes asked Apr 3

3 votes

1 answer

13

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 12
Consider two matrices $\mathrm{A}_{6 \times 3}$ and $\mathrm{B}_{3 \times 6}$, the non zero eigenvalues(EVs) of matrix $A B$ are $3,2,7,8$; see the following statements $\mathrm{S} 2$ : The EVs of BA must be all 0 for the ... false Both $\mathrm{S} 1$ and $\mathrm{S} 2$ are true Neither $\mathrm{S} 1$ nor $\mathrm{S} 2$ is true

Consider two matrices $\mathrm{A}_{6 \times 3}$ and $\mathrm{B}_{3 \times 6}$, the non zero eigenvalues(EVs) of matrix $A B$ are $3,2,7,8$; see the following statements$\...

GO Classes

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GO Classes asked Apr 3

2 votes

1 answer

14

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 13
Which of the followings(s) is/are TRUE ? If a system of linear equations has no free variables, then it has a unique solution. If an augmented matrix $[A \mid b]$ is transformed into $[C \mid d]$ by elementary row operations, ... equation $A x=b$ has $a$ unique solution for every $b$ in the set of real numbers of dimension $m$.

Which of the followings(s) is/are TRUE ?If a system of linear equations has no free variables, then it has a unique solution.If an augmented matrix $[A \mid b]$ is transf...

GO Classes

52 views

GO Classes asked Apr 3

4 votes

1 answer

15

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 14
Which of the following(s) is/are TRUE ? If none of the vectors in the set $S=\left\{\vec{v}_1, \vec{v}_2, \vec{v}_3\right\}$ in $\mathbb{R}^3$ is a multiple of one of the other vectors, then $S$ ... $\vec{v}_2$. Then $\left\{\vec{v}_1, \vec{v}_2, \vec{v}_3\right\}$ is linearly independent.

Which of the following(s) is/are TRUE ?If none of the vectors in the set $S=\left\{\vec{v}_1, \vec{v}_2, \vec{v}_3\right\}$ in $\mathbb{R}^3$ is a multiple of one of the ...

GO Classes

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GO Classes asked Apr 3

1 votes

1 answer

16

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 15
Consider the following two statements: $S 1$. If $A B=I$, then $A$ is invertible. $S 2$. If $A$ is a $3 \times 3$ matrix and the equation $A \vec{x}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has a unique ... $S 2$ is false $\mathrm{S} 1$ is false but $\mathrm{S} 2$ is true Both are true Both are false

Consider the following two statements:$S 1$. If $A B=I$, then $A$ is invertible.$S 2$. If $A$ is a $3 \times 3$ matrix and the equation $A \vec{x}=\left[\begin{array}{l}1...

GO Classes

98 views

GO Classes asked Apr 3

7 votes

1 answer

17

GO Classes Test Series 2024 | Mock GATE | Test 13 | Question: 13
If $A$ is a $3 \times 3$ matrix such that $A\left(\begin{array}{l}0 \\ 1 \\ 2\end{array}\right)=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$ ... $\left(\begin{array}{r}1 \\ -1 \\ 0\end{array}\right)$ $\left(\begin{array}{r}9 \\ 10 \\ 11\end{array}\right)$

If $A$ is a $3 \times 3$ matrix such that $A\left(\begin{array}{l}0 \\ 1 \\ 2\end{array}\right)=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$ and $A\left(\begin{ar...

GO Classes

581 views

GO Classes asked Jan 28

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0 answers

18

Linear Transformation of Matrix

Debargha Mitra Roy

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Debargha Mitra Roy asked Jan 12

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19

Diagonalization of Matrix

Debargha Mitra Roy

63 views

Debargha Mitra Roy asked Jan 11

0 votes

0 answers

20

Diagonalization of Matrix - Orthogonal Transformation
Consider a symmetric matrix $M=\begin{bmatrix} \frac{1}{3} & 0 & \frac{2}{3}\\ 0&1 &0 \\ \frac{2}{3}&0 & \frac{1}{3} \end{bmatrix}$. An orthogonal matrix $O$ which can diagonalize this matrix by an orthogonal transformation $O^TMO$ is given by $O = $ ______

Consider a symmetric matrix $M=\begin{bmatrix} \frac{1}{3} & 0 & \frac{2}{3}\\ 0&1 &0 \\ \frac{2}{3}&0 & \frac{1}{3} \end{bmatrix}$. An orthogonal matrix $O$ which can di...

Debargha Mitra Roy

105 views

Debargha Mitra Roy asked Jan 11

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21

GATE 2022 | MATHS | Q-25
Consider the linear system of equations $Ax = b$ with \[ A = \begin{bmatrix} 3 & 1 & 1 \\ 1 & 4 & 1 \\ 2 & 0 & 3 \\ \end{bmatrix} \] and \[ b = \begin{bmatrix} 2 \\ 3 \\ 4 \\ \end ... for any initial vector. (C) The Gauss-Seidel iterative method converges for any initial vector. (D) The spectral radius of the Jacobi iterative matrix is less than 1.

Consider the linear system of equations $Ax = b$ with\[ A =\begin{bmatrix}3 & 1 & 1 \\1 & 4 & 1 \\2 & 0 & 3 \\\end{bmatrix}\]and\[ b =\begin{bmatrix}2 \\3 \\4 \\\end{bm...

rajveer43

80 views

rajveer43 asked Jan 10

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0 answers

22

Linear Algebra, Eigen Vales & Eigen Vectors
$If \ A = \begin{pmatrix} 1&1 \\ 1&0 \end{pmatrix},\ \alpha M_1+\beta M_2+\gamma M_3,\ where \ M_1 = L_{2x2},\ M_2 = \begin{pmatrix} 0&1 \\ 1&1 \end{pmatrix}\ and \ M_3 = \begin{pmatrix} 1&1 \\ 1&1 \end{pmatrix} \ then \ -$ ... $\alpha = 1,\ \beta = -1,\ \gamma = 2$ D. $\alpha = -1,\ \beta = 1,\ \gamma = 2$

$If \ A = \begin{pmatrix} 1&1 \\ 1&0 \end{pmatrix},\ \alpha M_1+\beta M_2+\gamma M_3,\ where \ M_1 = L_{2x2},\ M_2 = \begin{pmatrix} 0&1 \\ 1&1 \end{pmatrix}\ and \ M_3 =...

Debargha Mitra Roy

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Debargha Mitra Roy asked Jan 8

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0 answers

23

Find the basic feasible solutions
Find the basic feasible solutions of the system of equations :- $x_1+x_2+x_3=8,$ $3x_1+2x_2=18,$ $x_1,x_2,x_3≥ 0$

Find the basic feasible solutions of the system of equations :-$x_1+x_2+x_3=8,$$3x_1+2x_2=18,$$x_1,x_2,x_3≥ 0$

Debargha Mitra Roy

146 views

Debargha Mitra Roy asked Dec 11, 2023

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