Syllabus: Matrices, determinants, System of linear equations, Eigenvalues and eigenvectors, LU decomposition.

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2024-1} & \textbf{2024-2} & \textbf{2023} & \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} &1&0&2& 1 &0&1&0&0.83&2
\\\hline\textbf{2 Marks Count} &1&1&0& 2 &1&1&0&1&2
\\\hline\textbf{Total Marks} &3&3&2& 5 &2&3&\bf{2}&\bf{3}&\bf{5}\\\hline
\end{array}}}$$

Recent questions in Linear Algebra

0 votes

1 answer

1

IIT Madras MS DSAI Written Test 2024
If the determinant and sum of eigen values of a 2 x 2 matrix are -1 and 0 then, what can you say about the rank of the given matrix? a) rank is 0 b) rank is 1 c) Insufficient information d) rank is 2

If the determinant and sum of eigen values of a 2 x 2 matrix are -1 and 0 then, what can you say about the rank of the given matrix?a) rank is 0b) rank is 1c) Insufficien...

102 views

harshrajhrj asked 6 days ago

0 votes

1 answer

2

IIT Madras MS DSAI Written Test 2024
Given, that the eigen values of a 2 x 2 matrix are -1,1 and its singular values are 1,0. What is the rank of the matrix? a) rank is 0 b) rank is 1 c) Such a matrix can't exist d) rank is 2

Given, that the eigen values of a 2 x 2 matrix are -1,1 and its singular values are 1,0. What is the rank of the matrix?a) rank is 0b) rank is 1c) Such a matrix can't exi...

70 views

harshrajhrj asked 6 days ago

4 votes

1 answer

3

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 1
Let $T_{1}, T_{2}: R^{5} \rightarrow R^{3}$ be linear transformations s.t $\operatorname{rank}\left(T_{1}\right)=3$ and nullity $\left(T_{2}\right)=3$. Let $T_{3}: R^{3} \rightarrow R^{3}$ be linear transformation s.t $T_{3}\left(T_{1}\right)=T_{2}$. Then find rank of $T_{3}$

Let $T_{1}, T_{2}: R^{5} \rightarrow R^{3}$ be linear transformations s.t $\operatorname{rank}\left(T_{1}\right)=3$ and nullity $\left(T_{2}\right)=3$. Let $T_{3}: R^{3} ...

87 views

GO Classes asked Apr 11

4 votes

1 answer

4

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 2
Suppose that $\left\{\mathbf{v}_{\mathbf{1}}, \mathbf{v}_{\mathbf{2}}, \mathbf{v}_{\mathbf{3}}\right\}$ is a linearly independent set of vectors in $\mathbb{R}^{6}$ ... is linearly independent $\left\{\mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{w}\right\}$ is linearly independent

Suppose that $\left\{\mathbf{v}_{\mathbf{1}}, \mathbf{v}_{\mathbf{2}}, \mathbf{v}_{\mathbf{3}}\right\}$ is a linearly independent set of vectors in $\mathbb{R}^{6}$.Furth...

93 views

GO Classes asked Apr 11

3 votes

1 answer

5

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...

78 views

GO Classes asked Apr 11

2 votes

1 answer

6

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 4
Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\}$ and $\mathcal{C}=\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \mathbf{c}_{3}\right\}$ be two bases of $\mathbb{R}^{3}$, and ... $\left[\begin{array}{r}3 \\ -1 \\ 1\end{array}\right]$

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\}$ and $\mathcal{C}=\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \mathbf{c}_{3}\right\}$ be two bas...

71 views

GO Classes asked Apr 11

5 votes

2 answers

7

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 5
Consider the linear map $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined by $ T(x, y)=(x-y, x-2 y), \text { for } x, y \in \mathbb{R} $ Let $\mathcal{E}$ be the standard basis for $\mathbb{R}^{2}$ ... $\left(\begin{array}{ll}0 & -1 \\ 1 & -1\end{array}\right)$

Consider the linear map $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined by$$T(x, y)=(x-y, x-2 y), \text { for } x, y \in \mathbb{R}$$Let $\mathcal{E}$ be the stand...

80 views

GO Classes asked Apr 11

3 votes

1 answer

8

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 6
Suppose that ...

Suppose that$$\left[\left[\begin{array}{l}1 \\2\end{array}\right]\right]_{\mathcal{B}}=\left[\begin{array}{l}3 \\4\end{array}\right] \text { and }\left[\left[\begin{array...

67 views

GO Classes asked Apr 11

4 votes

1 answer

9

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 7
Consider a $4 \times 4$ matrix $A$ and a vector $\mathbf{v} \in \mathbb{R}^{4}$ such that $A^{4} \mathbf{v}=\mathbf{0}$ but $A^{3} \mathbf{v} \neq \mathbf{0}$ ... $\mathbb{R}^{4}$. $\mathcal{B}$ is a basis of $\mathbb{R}^{4}$. $\mathcal{B}$ is not linearly independent.

Consider a $4 \times 4$ matrix $A$ and a vector $\mathbf{v} \in \mathbb{R}^{4}$ such that $A^{4} \mathbf{v}=\mathbf{0}$ but $A^{3} \mathbf{v} \neq \mathbf{0}$.Set $\mathc...

85 views

GO Classes asked Apr 11

1 votes

1 answer

10

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 8
Consider a linear system $A \mathbf{x}=\mathbf{b}$, where $A$ is a $3 \times 4$ matrix with $\operatorname{Rank}(A)=2$ ... unique solution. (i) infinitely many solutions, (ii) no solution. (i) infinitely many solutions, (ii) unique solution.

Consider a linear system $A \mathbf{x}=\mathbf{b}$, where $A$ is a $3 \times 4$ matrix with $\operatorname{Rank}(A)=2$.How many solutions does this system have if $(i) \o...

62 views

GO Classes asked Apr 11

2 votes

1 answer

11

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 9
Suppose that $a \neq 0$ and $a \neq b$. Which equation below is the equation relating $a, b$ and $c$ ... $4 a-3 b+c \neq 0$ $3 a-4 b-c \neq 0$ $4 a-3 b-c \neq 0$

Suppose that $a \neq 0$ and $a \neq b$. Which equation below is the equation relating $a, b$ and $c$ so that the vectors$$\left[\begin{array}{l}1 \\1 \\a\end{array}\right...

63 views

GO Classes asked Apr 11

1 votes

1 answer

12

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 10
Let $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ ... $\left(\begin{array}{lll}2 & 1 & 3 \\ 6 & 3 & 9\end{array}\right)$

Let $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ be a linear transformation such that $T\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right]=\left[\begin{array}{l}2 \\ 6...

64 views

GO Classes asked Apr 11

3 votes

1 answer

13

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 11
Which of the following statements are true? There exists a $3 \times 3$ matrix $A$ and vectors $b, c \in \mathbb{R}^{3}$ such that the linear system $A x=b$ has a unique solution but $A x=c$ has infinitely ... $n$, then the column space of $A$ is equal to the column space of $B$.

Which of the following statements are true?There exists a $3 \times 3$ matrix $A$ and vectors $b, c \in \mathbb{R}^{3}$ such that the linear system $A x=b$ has a unique s...

59 views

GO Classes asked Apr 11

3 votes

2 answers

14

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 12
Consider two statements S1 and S2. S1: If $\left\{v_{1}, \ldots, v_{n}\right\}$ are linearly INDEPENDENT vectors in $V$, then $\left\{T\left(v_{1}\right), \ldots, T\left(v_{n}\right)\right\}$ are linearly ... $\mathrm{S} 2$ is true. Both S1 and S2 are true. Both S1 and S2 are false.

Consider two statements S1 and S2.S1: If $\left\{v_{1}, \ldots, v_{n}\right\}$ are linearly INDEPENDENT vectors in $V$, then $\left\{T\left(v_{1}\right), \ldots, T\left(v...

76 views

GO Classes asked Apr 11

5 votes

2 answers

15

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 13
Suppose $A$ is a $4 \times 3$ matrix and $B$ is a $3 \times 2$ matrix, and let $T$ be the matrix transformation $T(x)=A B x$. Which of the following must be true? The column space of $A B$ ... space of $A$. $T$ has domain $\mathbf{R}^{2}$ and codomain $\mathbf{R}^{4}$. $T$ cannot be onto.

Suppose $A$ is a $4 \times 3$ matrix and $B$ is a $3 \times 2$ matrix, and let $T$ be the matrix transformation $T(x)=A B x$. Which of the following must be true?The colu...

72 views

GO Classes asked Apr 11

2 votes

1 answer

16

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 14
Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$ ... the matrix equation $A x=0$ has infinitely many solutions, then $\operatorname{rank}(A) \leq 4$.

Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$. Which of the following statements are true?$\op...

71 views

GO Classes asked Apr 11

2 votes

1 answer

17

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 15
The matrix $ \left(\begin{array}{rr} -2 & 11 \\ 4 & 2 \end{array}\right) $ represents a linear transformation $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ ... $\left(\begin{array}{rr}-2 & 11 \\ 4 & 2\end{array}\right)$

The matrix$$\left(\begin{array}{rr}-2 & 11 \\4 & 2\end{array}\right)$$represents a linear transformation $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ with respect to th...

60 views

GO Classes asked Apr 11

4 votes

1 answer

18

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 16
Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$ ... the matrix equation $A x=0$ has infinitely many solutions, then $\operatorname{rank}(A) \leq 4$.

Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$. Which of the following statements are true?$\op...

69 views

GO Classes asked Apr 11

6 votes

2 answers

19

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...

177 views

GO Classes asked Apr 3

7 votes

3 answers

20

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...

205 views

GO Classes asked Apr 3

Page:

1
2
3
4
5
6
...
58
next »