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

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151

Pyq
If a square matrix of order 100 has exactly 15 distinct eigen values, the degree of the minimal polynomial is (a) At least 15 (c) Always 15 (b) At most 15 (d) Exactly 100

If a square matrix of order 100 has exactly 15 distinct eigen values, the degree of the minimal polynomial is (a) At least 15 (c) Always 15 (b) At most 15 (d) Exactly 100...

Gaurav Kadyan

227 views

Gaurav Kadyan asked Nov 10, 2023

10 votes

2 answers

152

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...

GO Classes

479 views

GO Classes asked Mar 29, 2023

1 votes

1 answer

153

ME: GATE-2005
A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. The highest possible rank of A is:____________

A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. Thehighest possible rank of A is:____________

Hira Thakur

4.3k views

Hira Thakur asked Nov 28, 2017

41 votes

3 answers

154

GATE CSE 2014 Set 3 | Question: 4
Which one of the following statements is TRUE about every $n \times n$ matrix with only real eigenvalues? If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is ... eigenvalues are positive. If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.

Which one of the following statements is TRUE about every $n \times n$ matrix with only real eigenvalues?If the trace of the matrix is positive and the determinant of the...

go_editor

11.3k views

go_editor asked Sep 28, 2014

18 votes

2 answers

155

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{...

GO Classes

1.1k views

GO Classes asked Apr 5, 2023

16 votes

3 answers

156

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 22
A three-by-three matrix $B$ is known to have eigenvalues $0,1$ and $2.$ This information is enough to find which one of these (give the answers where possible): The rank of $B$ The determinant of $B^T B$ The eigenvalues of $B^T B$ The eigenvalues of $\left(B^2+I\right)^{-1}$

A three-by-three matrix $B$ is known to have eigenvalues $0,1$ and $2.$ This information is enough to find which one of these (give the answers where possible):The rank o...

GO Classes

1.3k views

GO Classes asked Mar 29, 2023

1 votes

1 answer

157

Self doubt: what are eigen values of resultant matrix when add two matrices?
if eigen values of matrix A = a,b,c if eigen values of matrix B = x,y,z then is it holds every time that eigen value of A+B = a+x, b+y, c+z ? please give some reference to you answer, thankyou.

if eigen values of matrix A = a,b,cif eigen values of matrix B = x,y,zthen is it holds every time that eigen value of A+B = a+x, b+y, c+z ?please give some reference to y...

jaswanth431

284 views

jaswanth431 asked Dec 12, 2021

17 votes

1 answer

158

GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 19
Suppose that we are solving $A x=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$ In each of the option below, a complete solution $x$ is proposed. Which of the following could possibly be the solution for above system of linear ... $\alpha \in \mathbb{R}$

Suppose that we are solving $A x=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$In each of the option below, a complete solution $x$ is proposed.Which of the followi...

GO Classes

1.1k views

GO Classes asked Apr 5, 2023

0 votes

1 answer

159

practice book gate
the eigen values of the matrix are (a) (a+1), 0 (b) a, 0 (c) (a-1), 0 (d) 0, 0

the eigen values of the matrix are (a) (a+1), 0(b) a, 0(c) (a-1), 0(d) 0, 0

akshay_123

290 views

akshay_123 asked Sep 7, 2023

0 votes

1 answer

160

practice book gate
The number of linearly independent solutions of the system of equations is equal to A- 1 B- 2 C-3 D- 0

The number of linearly independent solutions of the system of equationsis equal toA- 1B- 2C-3D- 0

akshay_123

411 views

akshay_123 asked Sep 8, 2023

21 votes

2 answers

161

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 ...

GO Classes

831 views

GO Classes asked Mar 29, 2023

15 votes

3 answers

162

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 26
Given a matrix $A\left(m \times n\right)$ and $A x=b$. Consider below statements : $\text{S}: m<n$ $\mathrm{P}:\; A$ has $m \;\mathrm{Linearly ~Independent}$ columns $\text{R}:$ There is a solution for every $b$ ... $\text{S} \rightarrow \sim \text{P}$ $(S$ and $\text{P}) \rightarrow \text{R}$

Given a matrix $A\left(m \times n\right)$ and $A x=b$. Consider below statements :$\text{S}: m<n$$\mathrm{P}:\; A$ has $m \;\mathrm{Linearly ~Independent}$ columns$\text{...

GO Classes

1.0k views

GO Classes asked Mar 29, 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}...

GO Classes

909 views

GO Classes asked Mar 29, 2023

16 votes

2 answers

164

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...

GO Classes

1.0k views

GO Classes asked Mar 29, 2023

26 votes

3 answers

165

GATE CSE 1994 | Question: 1.2
Let $A$ and $B$ be real symmetric matrices of size $n \times n$. Then which one of the following is true? $AA'=I$ $A=A^{-1}$ $AB=BA$ $(AB)'=BA$

Let $A$ and $B$ be real symmetric matrices of size $n \times n$. Then which one of the following is true?$AA'=I$$A=A^{-1}$$AB=BA$$(AB)'=BA$

Kathleen

7.8k views

Kathleen asked Oct 4, 2014

23 votes

1 answer

166

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 27
Consider the following set of (column) vectors: $X=\left\{\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right] \in \mathbb{R}^3 \mid 2 x_1+3 x_2-x_3=0\right\}.$ Which of the following statements are true? Every ... the correct option. Only I is true. Only II is true. Both I and II are true. Neither I, nor II are true.

Consider the following set of (column) vectors:$$X=\left\{\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right] \in \mathbb{R}^3 \mid 2 x_1+3 x_2-x_3=0\right\}.$$Whic...

GO Classes

701 views

GO Classes asked Mar 29, 2023

26 votes

5 answers

167

GATE CSE 1998 | Question: 2.2
Consider the following determinant $\Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}$ Which of the following is a factor of $\Delta$? $a+b$ $a-b$ $a+b+c$ $abc$

Consider the following determinant $\Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}$Which of the following is a factor of $\Delta$?$a+b$$a-b...

Kathleen

7.4k views

Kathleen asked Sep 25, 2014

13 votes

2 answers

168

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 ...

GO Classes

925 views

GO Classes asked Apr 5, 2023

18 votes

2 answers

169

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 15
$\mathrm{S} 1:$ A set of two vectors in $\mathbb{R}^n$ is Linealy dependent if at least one vector is a multiple of the other. $\text{S2}:$ A set of $n$ vectors in $\mathbb{R}^n$ is Linealy independent if and ... $\text{S} 1$ is incorrect $\text{S} 1$ and $\text{S} 2$ both are incorrect

$\mathrm{S} 1:$ A set of two vectors in $\mathbb{R}^n$ is Linealy dependent if at least one vector is a multiple of the other.$\text{S2}:$ A set of $n$ vectors in $\mathb...

GO Classes

967 views

GO Classes asked Mar 14, 2023

13 votes

3 answers

170

GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 9
Consider two statements below - Statement $1: $ If $A$ is invertible and $\lambda$ is an eigenvalue of $A$, then $\frac{1}{\lambda}$ is an eigenvalue of $A^{-1}$. Statement $2:$ Let $A$ be a real skew- ... true but Statement $2$ is false Statement $2$ is true but Statement $1$ is false Both statements are true Both statements are false

Consider two statements below -Statement $1: $ If $A$ is invertible and $\lambda$ is an eigenvalue of $A$, then $\frac{1}{\lambda}$ is an eigenvalue of $A^{-1}$.Statement...

GO Classes

779 views

GO Classes asked Apr 5, 2023

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