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Recent questions tagged linear-algebra

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1

GATE Electrical 2021 | Question: 1
Let $p$ and $q$ be real numbers such that $p^{2}+q^{2}=1$ . The eigenvalues of the matrix $\begin{bmatrix} p & q\\ q& -p \end{bmatrix}$are $1$ and $1$ $1$ and $-1$ $j$ and $-j$ $pq$ and $-pq$

Arjun asked in Linear Algebra Feb 20, 2021

by Arjun

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2

GATE Electrical 2021 | Question: 38
Let $A$ be a $10\times10$ matrix such that $A^{5}$ is a null matrix, and let $I$ be the $10\times10$ identity matrix. The determinant of $\text{A+I}$ is ___________________.

Arjun asked in Linear Algebra Feb 20, 2021

by Arjun

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3

GATE Electrical 2020 | Question: 42
The number of purely real elements in a lower triangular representation of the given $3\times 3$ ... $5$ $6$ $8$ $9$

go_editor asked in Linear Algebra Feb 28, 2020

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4

Gate2006-EE
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KUSHAGRA गुप्ता asked in Linear Algebra Sep 29, 2019

by KUSHAGRA गुप्ता

120 points

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5

GATE Electrical 2019 | Question: 2
$M$ is $2 \times 2$ matrix with eigenvalues $4$ and $9.$ The eigenvalues of $M^{2}$ are $4$ and $9$ $2$ and $3$ $-2$ and $-3$ $16$ and $81$

Arjun asked in Linear Algebra Feb 12, 2019

by Arjun

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6

GATE Electrical 2019 | Question: 24
The rank of the matrix, $M = \begin{bmatrix} 0 &1 &1 \\ 1& 0 &1 \\ 1& 1 & 0 \end{bmatrix}$, is ______________.

Arjun asked in Linear Algebra Feb 12, 2019

by Arjun

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7

GATE Electrical 2019 | Question: 26
Consider a $2\times 2$ matrix $M=\begin{bmatrix} v_1 & v_2 \end{bmatrix}$, where $v_1$ and $v_2$ are the column vectors. Suppose $M^{-1}=\begin{bmatrix} u_1^T \\ u_2^T \end{bmatrix}$, where $u_1^T$ and $u_2^T$ are ... True and Statement $2$ is false Statement $2$ is true and Statement $1$ is false Both the Statements are true Both the statements are false

Arjun asked in Linear Algebra Feb 12, 2019

by Arjun

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8

GATE Electrical 2012 | Question: 41
The state variable description of an LTI system is given by ... $a_1 = 0, \: a_2 \neq 0, \: a_3 = 0$ $a_1 \neq 0, \: a_2 \neq 0, \: a_3 = 0$

Andrijana3306 asked in Linear Algebra Mar 24, 2018

by Andrijana3306

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9

GATE Electrical 2012 | Question: 26
Given that $\textbf{A}= \begin{bmatrix} -5 & -3 \\ 2 & 0 \end{bmatrix}$ and $\textbf{I} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, the value of $A^3$ is $15 \: \textbf{A} + 12 \: \textbf{I}$ $19 \: \textbf{A} + 30 \: \textbf{I}$ $17 \: \textbf{A} + 15 \: \textbf{I}$ $17 \: \textbf{A} + 21 \: \textbf{I}$

Andrijana3306 asked in Linear Algebra Mar 24, 2018

by Andrijana3306

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GATE Electrical 2018 | Question: 44
Let $A= \begin{bmatrix} 1 & 0 & -1 \\ -1 & 2 & 0 \\ 0 & 0 & -2 \end{bmatrix}$ and $B=A^3-A^2-4A+5I$, where $I$ is the $3 \times 3$ identify matrix. The determinant of $B$ is _______ (up to $1$ decimal place).

Arjun asked in Linear Algebra Feb 19, 2018

by Arjun

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11

GATE Electrical 2018 | Question: 17
Consider a non-singular $2 \times 2$ square matrix $\textbf{A}$. If $\text{trace}(\textbf{A})=4$ and $\text{trace}(\textbf{A}^2)=5$, the determinant of the matrix $\textbf{A}$ is _________ (up to $1$ decimal place).

Arjun asked in Linear Algebra Feb 19, 2018

by Arjun

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12

GATE Electrical 2017 Set 2 | Question: 28
The eigenvalues of the matrix given below are $\begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & -3 & -4 \end{bmatrix}$ $(0, -1, -3)$ $(0, -2, -3)$ $(0, 2, 3)$ $(0, 1, 3)$

Arjun asked in Linear Algebra Feb 27, 2017

by Arjun

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13

GATE Electrical 2017 Set 1 | Question: 1
The matrix $A=\begin{bmatrix} \frac{3}{2} &0 & \frac{1}{2}\\ 0& -1 &0 \\ \frac{1}{2} & 0 & \frac{3}{2} \end{bmatrix}$ has three distinct eigenvalues and one of its eigenvectors is $\begin{bmatrix} 1\\ 0\\ 1 \end{bmatrix}$. ... $\begin{bmatrix} 1\\ 0\\ -1 \end{bmatrix}$ $\begin{bmatrix} 1\\ -1\\ 1 \end{bmatrix}$

Arjun asked in Linear Algebra Feb 27, 2017

by Arjun

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14

GATE Electrical 2013 | Question: 51
The state variable formulation of a system is given as $\begin{bmatrix} x^\cdot_1 \\ x^\cdot_2 \end{bmatrix}=\begin{bmatrix} -2 & 0\\ 0 & -1 \end{bmatrix}\begin{bmatrix} x_1\\ x_2 \end{bmatrix}+\begin{bmatrix} 1\\ 1 \end{bmatrix}u$ , $x_1(0)=0$ , $x_2(0)=0$ ... $1-\dfrac{1}{2}e^{-2t}-\dfrac{1}{2}e^{-t} \\$ $e^{-2t}-e^{-t} \\$ $1-e^{-t}$

piyag476 asked in Linear Algebra Feb 12, 2017

1.5k points

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15

GATE Electrical 2013 | Question: 50
The state variable formulation of a system is given as ... The system is controllable but not observable not controllable but observable both controllable and observable both not controllable and not observable

piyag476 asked in Linear Algebra Feb 12, 2017

1.5k points

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16

GATE Electrical 2013 | Question: 35
A matrix has eigenvalues $-1$ and $-2$. The corresponding eigenvectors are $\begin{bmatrix} 1\\-1 \end{bmatrix}$ and $\begin{bmatrix} 1\\-2 \end{bmatrix}$ respectibely. The matrix is $\begin{bmatrix} 1 & 1\\ -1 & -2 \end{bmatrix} \\$ ... $\begin{bmatrix} 0& 1\\ -2 & 3 \end{bmatrix}$

piyag476 asked in Linear Algebra Feb 12, 2017

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17

GATE Electrical 2013 | Question: 25
The equation$\begin{bmatrix} 2&-2 \\ 1& -1 \end{bmatrix}\begin{bmatrix} x_1\\ x_2 \end{bmatrix}=\begin{bmatrix} 0\\0 \end{bmatrix}$ has no solution only one solution $\begin{bmatrix} x1\\x2 \end{bmatrix}=\begin{bmatrix} 0\\0 \end{bmatrix}$ non-zero unique solution multiple solutions

piyag476 asked in Linear Algebra Feb 12, 2017

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18

GATE Electrical 2014 Set 3 | Question: 1
Two matrices $A$ and $B$ are given below: $A=\begin{vmatrix} p & q\\ r & s \end{vmatrix}$; $B=\begin{vmatrix} p^2+q^2 & pr+qs\\ pr+qs &r^2+s^2 \end{vmatrix}$ If the rank of matrix $A$ is $N$, then the rank of matrix $B$ is $N/2$ $N – 1$ $N$ $2N$

makhdoom ghaya asked in Linear Algebra Feb 12, 2017

by makhdoom ghaya

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19

GATE Electrical 2014 Set 2 | Question: 18
The state transition matrix for the system $\begin{bmatrix} \dot{x_1}\\ \dot{x_2} \end{bmatrix}=\begin{bmatrix} 1 & 0\\ 1 & 1 \end{bmatrix}\begin{bmatrix} x_1\\ x_2 \end{bmatrix}+\begin{bmatrix} 1\\ 1 \end{bmatrix}u$ ... $\begin{bmatrix} e^t &te^t \\ 0&e^t \end{bmatrix}$

makhdoom ghaya asked in Linear Algebra Feb 12, 2017

by makhdoom ghaya

9.3k points

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20

GATE Electrical 2014 Set 2 | Question: 1
Which one of the following statements is true for all real symmetric matrices? All the eigenvalues are real. All the eigenvalues are positive. All the eigenvalues are distinct. Sum of all the eigenvalues is zero.

makhdoom ghaya asked in Linear Algebra Feb 12, 2017

by makhdoom ghaya

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21

GATE Electrical 2014 Set 1 | Question: 46
A system matrix is given as follows. $A=\begin{bmatrix} 0 & 1 & -1\\ -6 & -11 &6 \\ -6& -11& 5 \end{bmatrix}$ The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is _______

makhdoom ghaya asked in Linear Algebra Feb 12, 2017

by makhdoom ghaya

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22

GATE Electrical 2015 Set 2 | Question: 2
We have a set of $3$ linear equations in $3$ unknowns. $'X \equiv Y'$ means $X$ and $Y$ are equivalent statements and $'X \not\equiv Y'$ means $X$ and $Y$ are not equivalent statements. P: There is a unique solution. Q: The equations are linearly ... $P \equiv Q \not\equiv R \equiv S$ $P\not\equiv Q \not\equiv R \not\equiv S$

makhdoom ghaya asked in Linear Algebra Feb 12, 2017

by makhdoom ghaya

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23

GATE Electrical 2015 Set 1 | Question: 26
The maximum value of "a" such that the matrix $\begin{pmatrix} -3&0&-2 \\ 1&-1&0 \\ 0&a&-2 \end{pmatrix}$ has three linearly independent real eigenvectors is $\dfrac{2}{3\sqrt{3}} \\$ $\dfrac{1}{3\sqrt{3}} \\$ $\dfrac{1+2\sqrt{3}}{3\sqrt{3}} \\$ $\dfrac{1+\sqrt{3}}{3\sqrt{3}}$

makhdoom ghaya asked in Linear Algebra Feb 12, 2017

by makhdoom ghaya

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24

GATE Electrical 2015 Set 1 | Question: 3
If the sum of the diagonal elements of a $2 \times 2$ matrix is $-6$, then the maximum possible value of determinant of the matrix is ________

makhdoom ghaya asked in Calculus Feb 12, 2017

by makhdoom ghaya

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25

GATE Electrical 2016 Set 2 | Question: 49
Consider a linear time invariant system $\dot{x}=Ax$ with initial condition $x(0)$ at $t=0$. Suppose $\alpha$ and $\beta$ are eigenvectors of $(2 \times 2)$ matrix $A$ corresponding to distinct eigenvalues $\lambda_{1}$ and $\lambda_{2}$ respectively. Then the ... $e^{\lambda_{2}t}\alpha$ $e^{\lambda_{1}t}\alpha+e^{\lambda_{2}t}\beta$

makhdoom ghaya asked in Linear Algebra Jan 30, 2017

by makhdoom ghaya

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26

GATE Electrical 2016 Set 2 | Question: 32
Let $P=\begin{bmatrix} 3&1 \\ 1 & 3 \end{bmatrix}$ Consider the set $S$ of all vectors $\begin{pmatrix} x\\ y \end{pmatrix}$ such that $a^{2}+b^{2}=1$ ... with major axis along $\begin{pmatrix} 1\\ 1 \end{pmatrix}$ An ellipse with minor axis along $\begin{pmatrix} 1\\ 1 \end{pmatrix}$

makhdoom ghaya asked in Linear Algebra Jan 30, 2017

by makhdoom ghaya

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27

GATE Electrical 2016 Set 2 | Question: 7
A $3 \times 3$ matrix $P$ is such that, $P^{3}=P$. Then the eigenvalues of $P$ ܲ are $1, 1, −1$ $1, 0.5 + ݆j0.866, 0.5 − ݆j0.866$ $1,−0.5 + ݆j0.866, −0.5 − ݆j0.866$ $0, 1, −1$

makhdoom ghaya asked in Linear Algebra Jan 30, 2017

by makhdoom ghaya

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28

GATE Electrical 2016 Set 1 | Question: 29
Let $A$ be a $4 \times 3$ real matrix with rank $2$. Which one of the following statement is TRUE? Rank of $A^{T} A$ is less than $2$. Rank of $A^{T} A$ is equal to $2$. Rank of $A^{T} A$ is greater than $2$. Rank of $A^{T} A$ can be any number between $1$ and $3$.

makhdoom ghaya asked in Linear Algebra Jan 30, 2017

by makhdoom ghaya

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29

GATE Electrical 2016 Set 1 | Question: 28
Let the eigenvalues of a $2 \times 2$ matrix $A$ be $1, -2$ with eigenvectors $x_{1}$ and $x_{2}$ respectively. Then the eigenvalues and eigenvectors of the matrix $A^{2}-3A+4I$ would, respectively, be $2, 14; x_{1}, x_{2}$ $2, 14; x_{1}+ x_{2}, x_{1} - x_{2}$ $2, 0; x_{1}, x_{2}$ $2, 0; x_{1}+ x_{2}, x_{1} - x_{2}$

makhdoom ghaya asked in Linear Algebra Jan 30, 2017

by makhdoom ghaya

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30

GATE Electrical 2016 Set 1 | Question: 2
Consider a $3 \times 3$ matrix with every element being equal to $1$. Its only non-zero eigenvalue is ________.

makhdoom ghaya asked in Linear Algebra Jan 30, 2017

by makhdoom ghaya

9.3k points

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