Recent questions tagged linear-algebra / GATE Overflow for GATE CSE

16 votes

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211

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}...

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GO Classes asked Mar 29, 2023

10 votes

3 answers

212

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

21 votes

2 answers

213

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 ...

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GO Classes asked Mar 29, 2023

18 votes

2 answers

214

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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GO Classes asked Mar 29, 2023

10 votes

2 answers

215

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...

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GO Classes asked Mar 29, 2023

18 votes

1 answer

216

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 6
Let $\vec{v}$ be an eigenvector of an invertible matrix $A$. Which of the following are necessarily true? $\vec{v}$ is an eigenvector of $A^{-1}$. $\vec{v}$ is an eigenvector of $A^2$. $\vec{v}$ is an eigenvector of $A+I$. $\vec{v}$ is an eigenvector of $A+2 I$.

Let $\vec{v}$ be an eigenvector of an invertible matrix $A$. Which of the following are necessarily true?$\vec{v}$ is an eigenvector of $A^{-1}$.$\vec{v}$ is an eigenvect...

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GO Classes asked Mar 29, 2023

32 votes

2 answers

217

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}=\...

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GO Classes asked Mar 29, 2023

14 votes

3 answers

218

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 8
Which of the following is/are TRUE? If the echelon form of an $m \times n$ matrix has a pivot in every column then $n \geq m$ If the echelon form of an $m \times n$ matrix $A$ has a pivot in every ... solution. If a system of $25$ linear equations in $13$ unknowns has at least one solution then it has infinitely many solutions.

Which of the following is/are TRUE?If the echelon form of an $m \times n$ matrix has a pivot in every column then $n \geq m$If the echelon form of an $m \times n$ matrix ...

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GO Classes asked Mar 29, 2023

18 votes

4 answers

219

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 9
$ \left[\begin{array}{ll} 3 & 1 \\ 1 & 0 \\ 2 & 5 \end{array}\right]\left[\begin{array}{lll} a & 1 & 0 \\ 2 & b & 1 \end{array}\right]=A_{3 \times 3} $ At what values of $(a, b), A_{3 \times 3}$ will be invertible? $(1,-1)$ $(-1,1)$ Both A and B None of the above

$$\left[\begin{array}{ll}3 & 1 \\1 & 0 \\2 & 5\end{array}\right]\left[\begin{array}{lll}a & 1 & 0 \\2 & b & 1\end{array}\right]=A_{3 \times 3}$$At what values of $(a, b),...

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GO Classes asked Mar 29, 2023

10 votes

2 answers

220

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...

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GO Classes asked Mar 29, 2023

9 votes

2 answers

221

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...

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705 views

GO Classes asked Mar 29, 2023

27 votes

4 answers

222

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 12
If $P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $\mathrm{A},$ and $\operatorname{det(A)}=4,$ then $\alpha$ equals to?

If $P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $\mathrm{A},$ and $\operatorname{det(A)}...

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GO Classes asked Mar 29, 2023

16 votes

2 answers

223

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...

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GO Classes asked Mar 29, 2023

28 votes

3 answers

224

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 14
Mark all the INCORRECT statements Let $A$ be the matrix of a rotation by angle $30$ degree. That is, for any vector $x,$ the angle between $x$ and $A x$ is always $30$ degree. Then $A$ ... $\lambda=1,2,3$, then $A$ is singular. If $A$ is a symmetric matrix, then all its eigenvectors are orthogonal.

Mark all the INCORRECT statementsLet $A$ be the matrix of a rotation by angle $30$ degree. That is, for any vector $x,$ the angle between $x$ and $A x$ is always $30$ deg...

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GO Classes asked Mar 29, 2023

19 votes

5 answers

225

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 15
Suppose $\mathbf{A}=\mathbf{B C}$, where $\mathbf{B}$ is a $4 \times 2$ matrix and $\mathbf{C}$ is a $2 \times 4$ matrix. Is $\mathbf{A}$ invertible? Yes, $\mathbf{A}$ is invertible. No, $\mathbf{A}$ is not invertible. Depends on $\mathbf{C}$ only. Depends on $\mathbf{B}$ and $\mathbf{C}$.

Suppose $\mathbf{A}=\mathbf{B C}$, where $\mathbf{B}$ is a $4 \times 2$ matrix and $\mathbf{C}$ is a $2 \times 4$ matrix. Is $\mathbf{A}$ invertible?Yes, $\mathbf{A}$ is ...

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GO Classes asked Mar 29, 2023

8 votes

1 answer

226

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 16
Let $A=\left[\begin{array}{cc}1 & 0 \\ -1 & 1 \\ k & 2\end{array}\right]$ and $b=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]$. For which value of $k$ does the system $A \mathbf{x}=b$ have a unique solution? There is no such value for $k$. $k=0$ $k=-1$ $k=1$

Let $A=\left[\begin{array}{cc}1 & 0 \\ -1 & 1 \\ k & 2\end{array}\right]$ and $b=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]$. For which value of $k$ does the sys...

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836 views

GO Classes asked Mar 29, 2023

10 votes

1 answer

227

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 17
Let $A$ be an $m \times n$-matrix and let $B$ be an $n \times m$-matrix. Then which of the following statement is not true for all such matrices? $B A$ is defined the columns of $A B$ are linear combinations of the columns of $B$ $A B$ is defined the columns of $A B$ are linear combinations of the columns of $A$

Let $A$ be an $m \times n$-matrix and let $B$ be an $n \times m$-matrix. Then which of the following statement is not true for all such matrices?$B A$ is definedthe colum...

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823 views

GO Classes asked Mar 29, 2023

6 votes

1 answer

228

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 18
Consider $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 0\end{array}\right]$. ... not of the columns of $B$. $\boldsymbol{v}$ is neither a linear combination of the columns of $A$ nor of the columns of $B$.

Consider $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 0\end{array}\right]$. Let ...

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591 views

GO Classes asked Mar 29, 2023

9 votes

1 answer

229

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 19
Let $A$ be a $2 \times 3$-matrix and $b$ a vector in $\mathbb{R}^2$. Consider the following two statements: $(\text{P}1)\; A$ has at most two pivots, $(\text{P2})$ Assuming $A x=b$ has a solution, then it ... $\text{P1}$ is correct. Statement $\text{P1}$ and Statement $\text{P2}$ are correct.

Let $A$ be a $2 \times 3$-matrix and $b$ a vector in $\mathbb{R}^2$.Consider the following two statements:$(\text{P}1)\; A$ has at most two pivots,$(\text{P2})$ Assuming ...

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561 views

GO Classes asked Mar 29, 2023

19 votes

3 answers

230

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 20
Suppose $A$ is $3$ by $4,$ and $A x=0$ ... $a, b, c,$ and $d?$ $a=-1$ $b=2$ $c=-1$ $d=1$

Suppose $A$ is $3$ by $4,$ and $A x=0$ has all solutions in the following form -$$\mathrm{x}=\mathrm{s}\left[\begin{array}{l}1 \\1 \\1 \\0\end{array}\right]+\mathrm{t}\le...

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GO Classes asked Mar 29, 2023

12 votes

2 answers

231

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 21
Consider a matrix $A$ of dimension $m \times n$ such that - $A x=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ has no solutions and $A x=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]$ has exactly one solution Which of the following CAN be true? $\operatorname{Rank}(A)=2$ $m=3$ $n=1$ $\operatorname{Rank}(A)=1$

Consider a matrix $A$ of dimension $m \times n$ such that -$A x=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ has no solutions and $A x=\left[\begin{array}{l}0 \\ ...

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GO Classes asked Mar 29, 2023

16 votes

3 answers

232

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...

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1.3k views

GO Classes asked Mar 29, 2023

26 votes

2 answers

233

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 23
Let $A$ be an $m \times n$-matrix. Consider the system of linear equations $A X=b$, which of the following statement is always true: Suppose $m>n$ then rank of augmented matrix $(A \mid b)$ can not larger than $m$. ... solution then $A X=0$ has unique solution. If $A X=b$ has a unique solution then $A$ has to be invertible.

Let $A$ be an $m \times n$-matrix. Consider the system of linear equations $A X=b$, which of the following statement is always true:Suppose $m>n$ then rank of augmented m...

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GO Classes asked Mar 29, 2023

24 votes

2 answers

234

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 24
Let $A$ be a $3 \times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1,$ and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1,$ ... ${\left[\begin{array}{c} 92 \\ -2 \\ 96 \end{array}\right]} $

Let $A$ be a $3 \times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1,$ and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and ...

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953 views

GO Classes asked Mar 29, 2023

17 votes

3 answers

235

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 25
Let $u \in \mathbb{R}^n$ be such that $u^T u=1$ and set $A=u u^T$. What will be the sum of all eigenvalues of $A?$

Let $u \in \mathbb{R}^n$ be such that $u^T u=1$ and set $A=u u^T$. What will be the sum of all eigenvalues of $A?$

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GO Classes asked Mar 29, 2023

15 votes

3 answers

236

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{...

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GO Classes asked Mar 29, 2023

23 votes

1 answer

237

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...

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GO Classes asked Mar 29, 2023

12 votes

2 answers

238

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 28
Choose the correct statements. If $b>\frac{2}{3}$, then $A=\left[\begin{array}{ll}2 & b \\ 3 & 1\end{array}\right]$ is invertible. If $P$ is an invertible matrix such that $B=P^{-1} A P,$ ... $m$ equations and $n$ variables, such that $m=n,$ always has a solution.

Choose the correct statements.If $b>\frac{2}{3}$, then $A=\left[\begin{array}{ll}2 & b \\ 3 & 1\end{array}\right]$ is invertible.If $P$ is an invertible matrix such that ...

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697 views

GO Classes asked Mar 29, 2023

7 votes

2 answers

239

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 29
Given matrix $A\left(m \times n\right)$ such that $A x=b.$ Consider below statements : $\text{S1}:$ If $b$ is Linearly Dependent on cols of matrix, then it need not always have a unique solution. $\text{S2}:$ If columns of matrix ... $\text{S2}$ is false $\mathrm{S} 2$ is true and $\mathrm{S} 1$ is false

Given matrix $A\left(m \times n\right)$ such that $A x=b.$ Consider below statements :$\text{S1}:$ If $b$ is Linearly Dependent on cols of matrix, then it need not always...

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GO Classes asked Mar 29, 2023

25 votes

2 answers

240

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 30
There exist a $3 \times 3$ real symmetric matrix $\text{S}$ ... is true but Statement $2$ is false Statement $1$ is false but Statement $2$ is true Both Statements are true Both Statements are false

There exist a $3 \times 3$ real symmetric matrix $\text{S}$ such that -$\text{Statement 1}: \text{S}\left(\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right)=\left(\begin{arr...

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GO Classes asked Mar 29, 2023

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