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

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}& \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{2020}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} & 1 &0&1&0&1&1&1&1&1&2&0&0.9&2
\\\hline\textbf{2 Marks Count} & 2 &1&1&1&1&1&2&1&0&0&0&1&2
\\\hline\textbf{Total Marks} & 5 &2&3&2&3&3&5&3&1&2&\bf{1}&\bf{2.9}&\bf{5}\\\hline
\end{array}}}$$

Hot questions in Linear Algebra

3 votes

2 answers

61

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 ...

73 views

GO Classes asked Apr 3

2 votes

1 answer

62

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...

19 views

GO Classes asked Apr 11

4 votes

1 answer

63

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 ...

68 views

GO Classes asked Apr 3

3 votes

1 answer

64

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...

25 views

GO Classes asked Apr 11

4 votes

1 answer

65

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 $B$?...

45 views

GO Classes asked Apr 3

3 votes

1 answer

66

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...

105 views

GO Classes asked Apr 3

2 votes

1 answer

67

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...

22 views

GO Classes asked Apr 11

3 votes

1 answer

68

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...

80 views

GO Classes asked Apr 3

1 votes

1 answer

69

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...

18 views

GO Classes asked Apr 11

1 votes

1 answer

70

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...

17 views

GO Classes asked Apr 11

11 votes

7 answers

71

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 8
$a\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4 \\ 5\end{array}\right]+b\left[\begin{array}{c}-1 \\ 2 \\ -3 \\ 4 \\ -5\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0 \\ 0 \\ 0\end{array}\right]$ How many number of pairs $(a, b)$ are there, that satisfy the above equation? $0$ $1$ Infinite $2$

$$a\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4 \\ 5\end{array}\right]+b\left[\begin{array}{c}-1 \\ 2 \\ -3 \\ 4 \\ -5\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0 \\ 0 ...

609 views

GO Classes asked Mar 14, 2023

2 votes

1 answer

72

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...

24 views

GO Classes asked Apr 11

2 votes

1 answer

73

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...

23 views

GO Classes asked Apr 11

3 votes

1 answer

74

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$\...

59 views

GO Classes asked Apr 3

3 votes

1 answer

75

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 ...

41 views

GO Classes asked Apr 3

38 votes

7 answers

76

GATE IT 2008 | Question: 29
If $M$ is a square matrix with a zero determinant, which of the following assertion (s) is (are) correct? S1: Each row of $M$ can be represented as a linear combination of the other rows S2: Each column of $M$ can be represented as a linear combination of the other columns S3 ... solution S4: $M$ has an inverse $S3$ and $S2$ $S1$ and $S4$ $S1$ and $S3$ $S1, S2$ and $S3$

If $M$ is a square matrix with a zero determinant, which of the following assertion (s) is (are) correct?S1: Each row of $M$ can be represented as a linear combination of...

9.6k views

Ishrat Jahan asked Oct 28, 2014

1 votes

2 answers

77

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...

58 views

GO Classes asked Apr 3

9 votes

2 answers

78

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

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 ...

509 views

GO Classes asked Jan 28

10 votes

3 answers

79

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...

690 views

GO Classes asked Mar 29, 2023

2 votes

1 answer

80

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...

52 views

GO Classes asked Apr 3

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