Recent questions tagged linear-algebra

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GATE2019 EC: 17
The number of distinct eigenvalues of the matrix $A=\begin{bmatrix} 2&2&3&3\\0&1&1&1\\0&0&3&3\\0&0&0&2 \end{bmatrix}$ is equal to ____________.

asked Feb 12, 2019 in Others by Arjun (1.4k points)

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GATE2019 EC: 20
Let $Z$ be an exponential random variable with mean $1$. That is, the cumulative distribution function of $Z$ is given by $F_{Z}(x)= \left\{\begin{matrix} 1-e^{-x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.$ Then $Pr\left(Z>2 \mid Z>1\right),$ rounded off to two decimal places, is equal to ___________.

asked Feb 12, 2019 in Others by Arjun (1.4k points)

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3

GATE2016-3-1
Consider a $2\times2$ sqaure matrix $\textbf{A}= \begin{bmatrix} \sigma &x\\ \omega &\sigma \end{bmatrix},$ where $x$ is unknown. If the eigen values of the matrix $\textbf{A}$ are $(\sigma + j\omega)$ and $(\sigma - j\omega)$, then $x$ is equal to $+j\omega$ $-j\omega$ $+\omega$ $-\omega$

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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GATE2016-3-34
The $z$-parameter matrix $\begin{bmatrix} z_{11} &z_{12}\\ z_{21} &z_{22} \end{bmatrix}$ for the two-port network shown is $\begin{bmatrix} 2 &-2\\-2 &2 \end{bmatrix} \\$ $\begin{bmatrix} 2 &2\\2 &2 \end{bmatrix} \\$ $\begin{bmatrix} 9 &-3\\6 &9 \end{bmatrix} \\$ $\begin{bmatrix} 9 &3\\6 &9 \end{bmatrix}$

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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5

GATE2016-2-1
The value of $x$ for which the matrix $A= \begin{bmatrix} 3& 2 &4 \\ 9& 7 & 13\\ -6&-4 &-9+x \end{bmatrix}$ has zero as an eigenvalue is ________

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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GATE2016-2-9
The $z$-parameter matrix for the two-port network shown is $\begin{bmatrix} 2j\omega &j\omega \\ j\omega & 3+2j\omega \end{bmatrix},$ where the entries are in $\Omega$. Suppose $Z_{b}\left ( j\omega \right )=R_{b}+j\omega .$ Then the value of $R_{b}$ (in $\Omega$) equals ________

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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7

GATE2016-2-29
The matrix $A=\begin{bmatrix} a & 0 &3 &7 \\ 2& 5&1 &3 \\ 0& 0& 2 &4 \\ 0&0 & 0 &b \end{bmatrix}$ has $\text{det}(A) = 100$ and $\text{trace}(A) = 14$. The value of $\mid a-b \mid$ is ________

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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GATE2016-1-6
Which one of the following is an eigen function of the class of all continuous-time, linear, time-invariant systems ($u(t)$ denotes the unit-step function)? $e^{j\omega_0t}u(t)$ $\cos(\omega_0t)$ $e^{j\omega_0t}$ $\sin(\omega_0t)$

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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GATE2016-1-27
A sequence $x[n]$ is specified as $\begin{bmatrix}x[n] \\x[n – 1]\end{bmatrix}=\begin{bmatrix}1&1\\1&0\end{bmatrix}^n\begin{bmatrix}1\\0\end{bmatrix}$,for $n \geq 2$. The initial conditions are $x[0] = 1$, $x[1] = 1$, and $x[n] = 0$ for $n < 0$. The value of $x[12]$ is _________

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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GATE2015-3-1
For $A = \begin{bmatrix} 1 &\tan x \\ -\tan x &1 \end{bmatrix},$ the determinant of $A^{T}A^{-1}$ is $\sec^{2}x$ $\cos 4x$ $1$ $0$

asked Mar 28, 2018 in Linear Algebra by Milicevic3306 (15.7k points)

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11

GATE2015-2-2
The value of $x$ for which all the eigen-values of the matrix given below are real is $\begin{bmatrix} 10&5+j &4 \\ x&20 &2 \\4 &2 &-10 \end{bmatrix}$ $5+j$ $5-j$ $1-5j$ $1+5j$

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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12

GATE2015-2-8
The $2$-port admittance matrix of the circuit shown is given by $\begin{bmatrix}0.3 &0.2 \\0.2 &0.3 \end{bmatrix}$ $\begin{bmatrix} 15&5 \\5 &15 \end{bmatrix}$ $\begin{bmatrix} 3.33&5 \\5 &3.33 \end{bmatrix}$ $\begin{bmatrix} 0.3&0.4 \\0.4 &0.3 \end{bmatrix}$

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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13

GATE2015-2-46
The state variable representation of a system is given as $\dot{x} = \begin{bmatrix} 0 &1 \\ 0 &-1 \end{bmatrix}\: ; x(0)=\begin{bmatrix} 1\\0 \end{bmatrix}$ $y=\begin{bmatrix} 0 &1 \end{bmatrix} x$ The response $y(t)$ is $\sin(t)$ $1-e^{t}$ $1-\cos(t)$ $0$

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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GATE2015-2-50
$\begin{Bmatrix} X_{n}\\ \end{Bmatrix}_{n=-\infty}^{n=\infty}$ is an independent and identically distributed (i.i.d.) random process with ܺ$X_{n}$ equally likely to be $+1$ or $−1.\:\:\begin{Bmatrix} Y_{n}\\ \end{Bmatrix}_{n=-\infty}^{n=\infty}$ is another ... $\begin{Bmatrix} Y_{n}\\ \end{Bmatrix}_{n=-\infty}^{n=\infty},$ denoted by $R_{Y}[k],$ is

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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15

GATE2015-1-1
Consider a system of linear equations: $x-2y+3z=-1, \\ x-3y+4z=1, \text{ and } \\ -2x+4y-6z=k.$ The value of $k$ for which the system has infinitely many solutions is ___________

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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16

GATE2015-1-5
The value of $p$ such that the vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is an eigenvector of the matrix $\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 \end{bmatrix}$ is _________.

asked Mar 28, 2018 in Others by Milicevic3306 (15.7k points)

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17

GATE2014-3-47
The state equation of a second-order linear system is given by $\dot{x}(t)=Ax(t), \:\:\:\:\:\:\:\:x(0)=x_{0}$ For $x_{0}= \begin{bmatrix} 1\\ -1 \end{bmatrix},$ $x(t)= \begin{bmatrix} e^{-t}\\ -e^{-t} \end{bmatrix},$ ... $\begin{bmatrix} 5e^{-t}-3e^{-2t}\\ -5e^{-t}+6e^{-2t} \end{bmatrix}$

asked Mar 26, 2018 in Others by Milicevic3306 (15.7k points)

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18

GATE2014-2-1
The determinant of matrix $A$ is $5$ and the determinant of matrix B is $40$. The determinant of matrix $AB$ is ________

asked Mar 26, 2018 in Others by Milicevic3306 (15.7k points)

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19

GATE2014-2-26
The system of linear equations $\begin{pmatrix} 2 & 1 & 3\\ 3&0 &1 \\ 1& 2 &5 \end{pmatrix} \begin{pmatrix} a\\ b\\ c \end{pmatrix} = \begin{pmatrix} 5\\ -4\\ 14 \end{pmatrix}$ has a unique solution infinitely many solutions no solution exactly two solutions

asked Mar 26, 2018 in Others by Milicevic3306 (15.7k points)

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20

GATE2014-1-4
A real $(4 \times 4)$ matrix $A$ satisfies the equation $A^{2} = I$, where $I$ is the $(4 \times 4)$ identity matrix. The positive eigen value of $A$ is ______.

asked Mar 26, 2018 in Linear Algebra by Milicevic3306 (15.7k points)

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21

GATE2013-55
The state diagram of a system is shown below. A system is described by the state-variable equations $\dot{X}= AX+Bu;\:\: y = CX+Du$ The state transition matrix $e^{At}$ of the system shown in the figure above is $\begin{bmatrix} e^{-t}& 0\\te^{-t} &e^{-t} \end{bmatrix}$ ... $\begin{bmatrix} e^{-t}&-te^{-t} \\ 0 &e^{-t} \end{bmatrix}$

asked Mar 26, 2018 in Linear Algebra by Milicevic3306 (15.7k points)

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22

GATE2013-27
Let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix. It is given that determinant $(I_{m} + AB) =$ determinant $(I_{n} + BA),$ where $I_{k}$ is the $k \times k$ ... $2$ $5$ $8$ $16$

asked Mar 26, 2018 in Linear Algebra by Milicevic3306 (15.7k points)

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23

GATE2013-19
The minimum eigenvalue of the following matrix is $\begin{bmatrix} 3& 5& 2\\5 &12 &7 \\2 &7 & 5\end{bmatrix}$ $0$ $1$ $2$ $3$

asked Mar 26, 2018 in Linear Algebra by Milicevic3306 (15.7k points)

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24

GATE2012-47
Given that $A=\begin{bmatrix} -5 &-3 \\ 2 &0\end{bmatrix}$ and $I=\begin{bmatrix} 1 & 0 \\ 0 &1\end{bmatrix}$, the value of $A^3$ is $15\:A+12\:I$ $19\:A+30\:I$ $17\:A+15\:I$ $17\:A+21\:I$

asked Mar 25, 2018 in Others by Milicevic3306 (15.7k points)

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25

GATE2018-22
Consider matrix $A=\begin{bmatrix} k & 2k\\ k^{2}-k & k^{2} \end{bmatrix}$ and vector $x=\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}.$ The number of distinct real value of $k$ for which the equation $Ax=0$ has infinitely many solutions is _________.

asked Feb 19, 2018 in Others by gatecse (1.4k points)

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26

GATE2018-11
Let $\text{M}$ be a real $4\times 4$ matrix. Consider the following statements: $S1: M $ has $4$ linearly independent eigenvectors. $S2: M$ has $4$ distinct eigenvalues. $S3: M$ is non-singular (invertible). Which one among the following is TRUE? $S1$ implies $S2$ $S1$ implies $S3$ $S2$ implies $S1$ $S3$ implies $S2$

asked Feb 19, 2018 in Others by gatecse (1.4k points)

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27

GATE2017 EC-2: 30
The minimum value of the function $f(x)=\frac{1}{3} x(x^2-3)$ in the interval $-100≤x≤100$ occurs at $x =$ ________.

asked Nov 23, 2017 in Linear Algebra by admin (2.8k points)

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28

GATE2017 EC-2: 4
The residues of a function $f(z)=\frac1{(z-4)(z+1)^3 }$ are $\frac{-1}{27}$ and $\frac{-1}{125} \\$ $\frac{1}{125}$ and $\frac{-1}{125} \\$ $\frac{-1}{27}$ and $\frac{1}{5} \\$ $\frac{1}{125}$and $\frac{-1}{5}$

asked Nov 23, 2017 in Linear Algebra by admin (2.8k points)

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29

GATE2017 EC-2: 1
The rank of the matrix $\begin{bmatrix} 1 & -1& 0& 0& 0& \\ 0& 0& 1& -1& 0& \\ 0& 1& -1& 0& 0& \\ -1& 0& 0& 0& 1& \\ 0& 0& 0& 1& -1& \end{bmatrix}$ is ________.

asked Nov 23, 2017 in Linear Algebra by admin (2.8k points)

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30

GATE2017 EC-1: 48
Which one of the following options correctly describes the locations of the roots of the equation $s^{4}+s^{2}+1=0$ on the complex plane? Four left half plane(LHP) roots One right half plane(RHP) root,one LHP root and two roots on the imaginary axis Two RHP roots and two LHP roots All four roots are on the imaginary axis

asked Nov 17, 2017 in Linear Algebra by admin (2.8k points)

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31

GATE2017 EC-1: 2
The rank of the matrix $\textbf{M} = \begin{bmatrix} 5&10&10 \\ 1 &0 &2 \\ 3&6&6 \end{bmatrix}$ is $0$ $1$ $2$ $3$

asked Nov 17, 2017 in Linear Algebra by admin (2.8k points)

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32

GATE2017 EC-1: 1
Consider the 5 $\times$ 5 matrix $\begin{bmatrix} 1&2&3&4&5\\ 5&1 &2& 3 &4\\ 4&5&1&2&3\\ 3&4&5&1&2\\ 2&3&4&5&1 \end{bmatrix}$ It is given that A has only one real eigenvalue. Then the real eigenvalue of A is $-2.5$ $0$ $15$ $25$

asked Nov 17, 2017 in Linear Algebra by admin (2.8k points)

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GATE2017 EC-1: 3
Consider the following statements about the linear dependence of the real valued function $y_{1}=1,y_{2}=x$ and $y_{3}=x^{2}$ over the field of real numbers. $y_{1},y_{2}$ and $y_{3} $ are linearly independent on $-1\leq x\leq 0$ ... Which one among the following is correct? Both I and II are true Both I and III are true Both II and IV are true Both III and IV are true

asked Nov 17, 2017 in Linear Algebra by admin (2.8k points)