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

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

Most answered questions in Linear Algebra

7 votes

2 answers

151

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 $\vec{p}$

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 sol...

564 views

GO Classes asked Feb 5

2 votes

2 answers

152

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 \(|\mathrm{M}^2 + 12M|\).

Consider the matrix\[\mathrm{M} = \begin{bmatrix}1 & 2 & 3 \\3 & 1 & 3 \\4 & 3 & 6\end{bmatrix}\]Find the value of \(|\mathrm{M}^2 + 12M|\).

542 views

GO Classes asked Feb 4

0 votes

2 answers

153

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.

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 ...

274 views

GO Classes asked Feb 4

9 votes

2 answers

154

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 ...

511 views

GO Classes asked Jan 28

7 votes

2 answers

155

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...

650 views

GO Classes asked Jan 21

0 votes

2 answers

156

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 \[ | \lambda - \mu | = \] _________

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 t...

104 views

rajveer43 asked Jan 11

0 votes

2 answers

157

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

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 ...

183 views

rajveer43 asked Jan 9

10 votes

2 answers

158

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...

826 views

GO Classes asked Apr 5, 2023

13 votes

2 answers

159

GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 8
Suppose that $A$ is a $3 \times 3$ real-symmetric matrix with eigenvalues $\lambda_1=1$, $\lambda_2=-1, \lambda_3=-2$, and corresponding eigenvectors $x_1, x_2, x_3.$ You are given that $x_1=\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)$ ... $x_3 =\left(\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right)$

Suppose that $A$ is a $3 \times 3$ real-symmetric matrix with eigenvalues $\lambda_1=1$, $\lambda_2=-1, \lambda_3=-2$, and corresponding eigenvectors $x_1, x_2, x_3.$You ...

922 views

GO Classes asked Apr 5, 2023

9 votes

2 answers

160

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$

678 views

GO Classes asked Apr 5, 2023

14 votes

2 answers

161

GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 18
In this problem, consider the $4 \times 4$ matrix $A$ whose columns are vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \mathbf{a}_4 \in \mathbb{R}^4$ ... a element in matrix $\text{A}$ at $\mathrm{i}^{\text {th }}$ row and $\mathrm{j}^{\text{th}}$ column.

In this problem, consider the $4 \times 4$ matrix $A$ whose columns are vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \mathbf{a}_4 \in \mathbb{R}^4$ :$$A=\left[\math...

755 views

GO Classes asked Apr 5, 2023

18 votes

2 answers

162

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{...

1.1k views

GO Classes asked Apr 5, 2023

16 votes

2 answers

163

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 1
Consider the vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ ... otherwise The vectors $\left\{\mathbf{v}_1, \mathbf{v}_2\right\}$ are linearly independent when $t=4$, and linearly dependent otherwise

Consider the vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ given by$$\mathbf{v}_1=\left(\begin{array}{l}2 \\t \\3\end{array}\right), \quad \mathbf{v}_2=\left(\begin{array}{l}...

892 views

GO Classes asked Mar 29, 2023

21 votes

2 answers

164

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 3
Let $A$ be a $n \times n$ matrix, and let $u, v, w$ be nonzero vectors in $\mathbf{R}^n$ which are distinct $(\text{so}\; u \neq v, u \neq w$, and $v \neq w).$ Suppose $A u=2 u, \quad A v=2 v, \quad A w=-w$. Which one of the following vectors must be an eigenvector of $A?$ $u-v$ $v-w$ $u-w$ none of the above

Let $A$ be a $n \times n$ matrix, and let $u, v, w$ be nonzero vectors in $\mathbf{R}^n$ which are distinct $(\text{so}\; u \neq v, u \neq w$, and $v \neq w).$Suppose $A ...

817 views

GO Classes asked Mar 29, 2023

18 votes

2 answers

165

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...

855 views

GO Classes asked Mar 29, 2023

10 votes

2 answers

166

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 5
Consider the system below, where $h$ and $k$ are real numbers. $ \begin{array}{r} x+3 y=2 \\ 3 x-h y=k \end{array} $ Find the values of $h$ and $k$ which give the system infinitely many solutions. $h=-9$ and $k \neq 6$ $h \neq-9$ and $k$ can be any real number $h=-9$ and $k=6$ System is inconsistent for all values of $h$ and $k$

Consider the system below, where $h$ and $k$ are real numbers.$$\begin{array}{r}x+3 y=2 \\3 x-h y=k\end{array}$$Find the values of $h$ and $k$ which give the system infin...

466 views

GO Classes asked Mar 29, 2023

32 votes

2 answers

167

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 7
Let $\mathrm{A}$ be a $3 \times 3$ matrix. Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$ ... $\mathrm{A}$.

Let $\mathrm{A}$ be a $3 \times 3$ matrix. Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have$$A \mathbf{x}=\...

1.5k views

GO Classes asked Mar 29, 2023

10 votes

2 answers

168

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 10
The given system has infinitely many solutions for $k=?$ $ \left[\begin{array}{ccc} 2 & 2 & -4 \\ 1 & 3 & -2 \\ -4 & k & 8 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 3 \\ 4 \\ -6 \end{array}\right] $

The given system has infinitely many solutions for $k=?$ $$\left[\begin{array}{ccc}2 & 2 & -4 \\1 & 3 & -2 \\-4 & k & 8\end{array}\right]\left[\begin{array}{l}x \\y \\z\e...

635 views

GO Classes asked Mar 29, 2023

9 votes

2 answers

169

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 11
In the echelon form of the augmented matrix $[A \mid b]:$ A single row of the form $(000 \ldots 0 \mid 0)$ ... have no solution. Only I is true. Only Il is true. Both I and II are true. Neither I, nor II are true.

In the echelon form of the augmented matrix $[A \mid b]:$ A single row of the form $(000 \ldots 0 \mid 0)$ is enough to conclude that the system will have infinitely many...

685 views

GO Classes asked Mar 29, 2023

16 votes

2 answers

170

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 13
Given the following matrix: $ A=\left[\begin{array}{lll} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{array}\right] $ Consider the following statements: $A^2-4 A-5 I=0$ (where $I$ is ... following options is correct? Only $I$ is true. Only II is true. Both I and II are true. Neither I, nor II are true.

Given the following matrix:$$A=\left[\begin{array}{lll}1 & 2 & 2 \\2 & 1 & 2 \\2 & 2 & 1\end{array}\right]$$Consider the following statements:$A^2-4 A-5 I=0$ (where $I$ i...

993 views

GO Classes asked Mar 29, 2023

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