Recent questions tagged rank-of-matrix / GATE Overflow for GATE CSE

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GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 3
Let the linear transformation $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ be defined by $T\left(x_{1}, x_{2}\right)=\left(x_{1}, x_{1}+x_{2}, x_{2}\right)$. Then the nullity of $T$ is: 0 1 2 3

Let the linear transformation $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ be defined by $T\left(x_{1}, x_{2}\right)=\left(x_{1}, x_{1}+x_{2}, x_{2}\right)$. Then the n...

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GO Classes asked Apr 11

1 votes

1 answer

2

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 ...

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GO Classes asked Apr 3

3 votes

1 answer

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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 ...

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GO Classes asked Apr 3

5 votes

1 answer

4

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.

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 i...

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GO Classes asked Feb 5

7 votes

2 answers

5

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 $11.$

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. W...

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GO Classes asked Jan 21

1 votes

0 answers

6

GO Classes Cs Test Series 2025 | Mixed Subjects | Full Length Test 1 | Question: 25
Let $\mathbf{x}=\left[\begin{array}{c}1 \\ -2 \\ 5\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}-7 \\ 4 \\ 2\end{array}\right]$ ... $A \mathbf{x}=\mathbf{v}$. Only I Only II I, II All of them

Let $\mathbf{x}=\left[\begin{array}{c}1 \\ -2 \\ 5\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{c}-7 \\ 4 \\ 2\end{array}\right]$. Which of the following statem...

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GO Classes asked Jun 10, 2023

1 votes

0 answers

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GO Classes Cs Test Series 2025 | Mixed Subjects | Full Length Test 1 | Question: 26
Suppose that $A$ is $n \times m$. Which of the following statements must be true? If $\operatorname{Rank}(A)=n$ and $m>n$ (i.e., $m \geq n+1$ ), then the system $A \mathbf{x}=\mathbf{0}$ has infinitely many ... has infinitely many solution. I, II, III II, IV I, III, V I, II, IV, V

Suppose that $A$ is $n \times m$. Which of the following statements must be true?If $\operatorname{Rank}(A)=n$ and $m>n$ (i.e., $m \geq n+1$ ), then the system $A \mathbf...

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GO Classes asked Jun 10, 2023

1 votes

0 answers

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GO Classes Cs Test Series 2025 | Mixed Subjects | Full Length Test 1 | Question: 30
A matrix $A$ ... $c$ the rank of $A$ must be $2$ must be $3$ must be $4$ can be $2, 3 \;\text{or}\; 4$

A matrix $A$ can be row reduced to the following echelon form$$\left[\begin{array}{lllll}1 & 1 & 3 & 2 & 1 \\0 & 1 & 2 & 1 & 3 \\0 & 0 & 0 & a & b \\0 & 0 & 0 & 0 & c\end...

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GO Classes asked Jun 10, 2023

1 votes

0 answers

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GO Classes Cs Test Series 2025 | Mixed Subjects | Full Length Test 1 | Question: 35
Suppose $A=\left[\begin{array}{llll}\mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n\end{array}\right]$ is an $n \times n$ invertible matrix, and $\mathbf{b}$ is a non-zero vector in $\mathbb{R}^n$ ... $A \mathbf{b}=\lambda \mathbf{b}$ for some constant $\lambda$, then $\lambda \neq 0$.

Suppose $A=\left[\begin{array}{llll}\mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n\end{array}\right]$ is an $n \times n$ invertible matrix, and $\mathbf{b}$ is a non...

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GO Classes asked Jun 10, 2023

2 votes

1 answer

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GO Classes Cs Test Series 2025 | Mixed Subjects | Full Length Test 1 | Question: 36
Consider the matrix $A=\left[\begin{array}{rrrr}1 & 3 & 1 & 0 \\ 0 & 2 & 4 & -2\end{array}\right]$ whose row echelon form is $\left[\begin{array}{llll}1 & 0 & a & b \\ 0 & 1 & c & d\end{array}\right]$. Find $a+b+c+d$ : $-7$ $-4$ $-3$ $-1$

Consider the matrix $A=\left[\begin{array}{rrrr}1 & 3 & 1 & 0 \\ 0 & 2 & 4 & -2\end{array}\right]$ whose row echelon form is $\left[\begin{array}{llll}1 & 0 & a & b \\ 0 ...

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GO Classes asked Jun 10, 2023

10 votes

2 answers

11

GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 5
Suppose that the characteristic polynomial of $\text{A}$ is $ p(\lambda)=\lambda(\lambda-2)(\lambda-3)^2. $ Which of the following can you determine from this information? The rank of $\text{A}$. Whether $\text{A}$ is symmetric. Whether $\text{A}$ is diagonalizable. The eigenvalues of $\text{A}$.

Suppose that the characteristic polynomial of $\text{A}$ is$$p(\lambda)=\lambda(\lambda-2)(\lambda-3)^2.$$Which of the following can you determine from this information?T...

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GO Classes asked Apr 5, 2023

9 votes

2 answers

12

GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 10
Let $A=\begin{pmatrix}1 & 2 & 1 & 0 & 0 \\ 1 & 2 & 2 & 2 & 3 \\ -1 & -2 & 0 & 2 & 3\end{pmatrix}$ What will be the $\text{rank(A)}?$ $1$ $2$ $3$ $5$

Let $A=\begin{pmatrix}1 & 2 & 1 & 0 & 0 \\ 1 & 2 & 2 & 2 & 3 \\ -1 & -2 & 0 & 2 & 3\end{pmatrix}$What will be the $\text{rank(A)}?$$1$$2$$3$$5$

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GO Classes asked Apr 5, 2023

18 votes

2 answers

13

GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 20
Consider Three matrices $A, B$ and $C$ ... $\alpha_1$, then: The Entries which are not shown in matrices are zeros. What is the rank of $B ?$

Consider Three matrices $A, B$ and $C$ such that -$$\underbrace{\left(\begin{array}{lllll}1 & 2 & 4 & 2 & 5 \\& 2 & 3 & 5 & 6 \\& & 3 & 4 & 3 \\& & & 4 & 3 \\& & & 5\end{...

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GO Classes asked Apr 5, 2023

11 votes

3 answers

14

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 2
Consider a matrix $A_{4 \times 5}$. Where all the solutions of $A x=0$ has the following form - $ \left[\begin{array}{c} 6 c-12 e \\ -4 c+10 e \\ c \\ -5 e \\ e \end{array}\right] $ What will be the rank of $A?$

Consider a matrix $A_{4 \times 5}$. Where all the solutions of $A x=0$ has the following form -$$\left[\begin{array}{c}6 c-12 e \\-4 c+10 e \\c \\-5 e \\e\end{array}\righ...

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GO Classes asked Mar 29, 2023

18 votes

2 answers

15

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 4
Consider a matrix $A_{n \times n}$ having the following characteristic equation - $ \lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3 $ What could be $\operatorname{rank}(\mathrm{A})?$ $6$ $7$ $8$ $9$

Consider a matrix $A_{n \times n}$ having the following characteristic equation -$$\lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3$$What could be $\operatorname{rank}(\mat...

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1.0k views

GO Classes asked Mar 29, 2023

3 votes

3 answers

16

Made Easy: Linear Algebra (MSQ)
Let $A$ be a $3$ x $3$ matrix with rank $2$. Then, $AX=0$ has The trivial solution $X=0$. One independent solution. Two independent solution. Three independent solution.

Let $A$ be a $3$ x $3$ matrix with rank $2$. Then, $AX=0$ hasThe trivial solution $X=0$.One independent solution.Two independent solution.Three independent solution.

DebRC

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DebRC asked Sep 19, 2022

1 votes

1 answer

17

TIFR CSE 2022 | Part B | Question: 14
Let $G$ be a directed graph (with no self-loops or parallel edges) with $n \geq 2$ vertices and $m$ edges. Consider the $n \times m$ incidence matrix $M$ of $G$, whose rows are indexed by the vertices of $G$ and the columns by the edges of $G$ ... . Then, what is the rank of $M?$ $m-1$ $m-n+1$ $\lceil m / 2\rceil$ $n-1$ $\lceil n / 2\rceil$

Let $G$ be a directed graph (with no self-loops or parallel edges) with $n \geq 2$ vertices and $m$ edges. Consider the $n \times m$ incidence matrix $M$ of $G$, whose ro...

admin

478 views

admin asked Sep 1, 2022

13 votes

1 answer

18

TIFR CSE 2022 | Part A | Question: 8
Let $A$ be the $(n+1) \times(n+1)$ matrix given below, where $n \geq 1$. For $i \leq n$, the $i$-th row of $A$ has every entry equal to $2i-1$ and the last row, i.e., the $(n+1)$-th row of $A$ has every entry equal to $-n^2$ ... $A$ has rank $n$ $A^2$ has rank $1$ All the eigenvalues of $A$ are distinct All the eigenvalues of $A$ are $0$ None of the above

Let $A$ be the $(n+1) \times(n+1)$ matrix given below, where $n \geq 1$. For $i \leq n$, the $i$-th row of $A$ has every entry equal to $2i-1$ and the last row, i.e., the...

admin

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admin asked Sep 1, 2022

3 votes

2 answers

19

GO Classes Test Series 2023 | Linear Algebra | Test | Question: 10
What could be the possible values of $\operatorname{rank}(\mathrm{A})$ as "$a$" varies: $ A=\left[\begin{array}{ccc} 1 & 2 & a\\ -2 & 4 a & 2\\ a & -2 & 1 \end{array}\right] $ For some value ... $a$, rank could be $1$ For some value of $a$, rank could be $2$ For some value of $a$, rank could be $3$

What could be the possible values of $\operatorname{rank}(\mathrm{A})$ as "$a$" varies:$$A=\left[\begin{array}{ccc}1 & 2 & a\\-2 & 4 a & 2\\a & -2 & 1\end{array}\right]$$...

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GO Classes asked Aug 14, 2022

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