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Recent questions and answers in Linear Algebra
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GATE EC 2021  Question: 36
A real $2\times2$ nonsingular matrix $A$ with repeated eigenvalue is given as $A=\begin{bmatrix} x & 3.0\\ 3.0 & 4.0 \end{bmatrix}$ where $x$ is a real positive number. The value of $x$ (rounded off to one decimal place) is ________________
asked
Feb 20
in
Linear Algebra
by
Arjun
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4.4k
points)

12
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GATE ECE 2020  Question: 26
Consider the following system of linear equations. $\begin{array}{llll} x_{1}+2x_{2}=b_{1} ; & 2x_{1}+4x_{2}=b_{2}; & 3x_{1}+7x_{2}=b_{3} ; & 3x_{1}+9x_{2}=b_{4} \end{array}$ Which one of the following conditions ensures that a solution exists for the above system? ... $b_{2}=2b_{1}$ and $3b_{1}6b_{3}+b_{4}=0$ $b_{3}=2b_{1}$ and $3b_{1}6b_{3}+b_{4}=0$
asked
Feb 13, 2020
in
Linear Algebra
by
jothee
(
1.8k
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26
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gate2020ec
linearalgebra
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0
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1
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3
GATE ECE 2013  Question: 19
The minimum eigenvalue of the following matrix is $\begin{bmatrix} 3& 5& 2\\5 &12 &7 \\2 &7 & 5\end{bmatrix}$ $0$ $1$ $2$ $3$
answered
Nov 26, 2019
in
Linear Algebra
by
Lakshman Patel RJIT
(
3.8k
points)

135
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gate2013ec
linearalgebra
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1
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4
GATE ECE 2019  Question: 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 ____________.
answered
Jul 23, 2019
in
Linear Algebra
by
yuviabhi
(
140
points)

59
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gate2019ec
numericalanswers
linearalgebra
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0
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0
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5
GATE ECE 2016 Set 3  Question: 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 $+j\omega$ $j\omega$ $+\omega$ $\omega$
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Mar 28, 2018
in
Linear Algebra
by
Milicevic3306
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15.8k
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27
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gate2016ec3
linearalgebra
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6
GATE ECE 2016 Set 2  Question: 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 ________
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Mar 28, 2018
in
Linear Algebra
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Milicevic3306
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18
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gate2016ec2
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7
GATE ECE 2016 Set 2  Question: 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 $\mid ab \mid$ is ________
asked
Mar 28, 2018
in
Linear Algebra
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Milicevic3306
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15.8k
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15
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gate2016ec2
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0
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8
GATE ECE 2016 Set 1  Question: 1
Let $M^4$= $I$,(where $I$ denotes the identity matrix) and $ M \neq I$, $M^2\neq I$ and $M^3\neq I$. Then,for any natural number $k$, $M^{1}$ equals: $M^{4k+1}$ $M^{4k+2}$ $M^{4k+3}$ $M^{4k}$
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Mar 28, 2018
in
Linear Algebra
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Milicevic3306
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15.8k
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15
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gate2016ec1
linearalgebra
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0
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9
GATE ECE 2016 Set 1  Question: 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
Linear Algebra
by
Milicevic3306
(
15.8k
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15
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gate2016ec1
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+1
vote
0
answers
10
GATE ECE 2015 Set 3  Question: 1
For $A = \begin{bmatrix} 1 &\tan x \\ \tan x &1 \end{bmatrix},$ the determinant of $A^{T}A^{1}$ is $\sec^{2}x$ $\cos 4x$ $1$ $0$
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Mar 28, 2018
in
Linear Algebra
by
Milicevic3306
(
15.8k
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13
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gate2015ec3
linearalgebra
matrices
determinant
0
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0
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11
GATE ECE 2015 Set 2  Question: 2
The value of $x$ for which all the eigenvalues of the matrix given below are real is $\begin{bmatrix} 10&5+j &4 \\ x&20 &2 \\4 &2 &10 \end{bmatrix}$ $5+j$ $5j$ $15j$ $1+5j$
asked
Mar 28, 2018
in
Linear Algebra
by
Milicevic3306
(
15.8k
points)

11
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gate2015ec2
linearalgebra
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0
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0
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12
GATE ECE 2015 Set 2  Question: 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 $\sin(t)$ $1e^{t}$ $1\cos(t)$ $0$
asked
Mar 28, 2018
in
Linear Algebra
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Milicevic3306
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15.8k
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23
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gate2015ec2
linearalgebra
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0
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13
GATE ECE 2015 Set 1  Question: 1
Consider a system of linear equations: $x2y+3z=1, \\ x3y+4z=1, \text{ and } \\ 2x+4y6z=k.$ The value of $k$ for which the system has infinitely many solutions is ___________
asked
Mar 28, 2018
in
Linear Algebra
by
Milicevic3306
(
15.8k
points)

20
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gate2015ec1
numericalanswers
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0
votes
0
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14
GATE ECE 2015 Set 1  Question: 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
Linear Algebra
by
Milicevic3306
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15.8k
points)

13
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gate2015ec1
numericalanswers
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0
votes
0
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15
GATE ECE 2015 Set 1  Question: 43
Two sequences $\begin{bmatrix}a, & b, & c \end{bmatrix}$ and $\begin{bmatrix}A, & B, & C \end{bmatrix}$ ... $\begin{bmatrix}p, & q, & r \end{bmatrix} = \begin{bmatrix} c, & b, & a \end{bmatrix}$
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Mar 28, 2018
in
Linear Algebra
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Milicevic3306
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15.8k
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19
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gate2015ec1
linearalgebra
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0
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0
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16
GATE ECE 2014 Set 4  Question: 46
The state transition matrix $\phi(t)$ of a system $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ is $\begin{bmatrix} t & 1 \\ 1 & 0 \end{bmatrix} \\$ ... $\begin{bmatrix} 0 & 1 \\ 1 & t \end{bmatrix} \\$ $\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}$
asked
Mar 26, 2018
in
Linear Algebra
by
Milicevic3306
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15.8k
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8
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gate2014ec4
linearalgebra
matrices
0
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0
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17
GATE ECE 2014 Set 3  Question: 27
Which one of the following statements is NOT true for a square matrix $A$? If $A$ is upper triangular, the eigenvalues of $A$ are the diagonal elements of it If $A$ is real symmetric, the eigenvalues of $A$ are always real and positive If $A$ ... $A$ are positive, all the eigenvalues of $A$ are also positive
asked
Mar 26, 2018
in
Linear Algebra
by
Milicevic3306
(
15.8k
points)

17
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gate2014ec3
linearalgebra
matrices
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0
votes
0
answers
18
GATE ECE 2014 Set 3  Question: 47
The state equation of a secondorder 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
Linear Algebra
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Milicevic3306
(
15.8k
points)

18
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gate2014ec3
linearalgebra
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0
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0
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19
GATE ECE 2014 Set 2  Question: 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
Linear Algebra
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Milicevic3306
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15.8k
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19
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gate2014ec2
numericalanswers
linearalgebra
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0
votes
0
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20
GATE ECE 2014 Set 2  Question: 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
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Mar 26, 2018
in
Linear Algebra
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Milicevic3306
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15.8k
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14
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gate2014ec2
linearalgebra
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0
votes
0
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21
GATE ECE 2014 Set 2  Question: 28
The maximum value of the determinant among all $2 \times 2$ real symmetric matrices with trace $14$ is __________.
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Mar 26, 2018
in
Linear Algebra
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Milicevic3306
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15.8k
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13
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gate2014ec2
numericalanswers
linearalgebra
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0
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0
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22
GATE ECE 2014 Set 1  Question: 1
For matrices of same dimension $M, N$ and scalar $c$, which one of these properties DOES NOT ALWAYS hold? $(M^{T})^{T} = M$ $(cM)^{T} = c(M)^{T}$ $(M+N)^{T} = M^{T} + N^{T}$ $MN = NM$
asked
Mar 26, 2018
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Linear Algebra
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Milicevic3306
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15.8k
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20
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gate2014ec1
linearalgebra
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0
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0
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23
GATE ECE 2014 Set 1  Question: 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
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Milicevic3306
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24
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gate2014ec1
linearalgebra
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0
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0
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24
GATE ECE 2014 Set 1  Question: 29
Consider the matrix ... $\alpha$ is a nonnegative real number. The value of $\alpha$ for which $\text{det(P)} = 0$ is _______.
asked
Mar 26, 2018
in
Linear Algebra
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Milicevic3306
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15.8k
points)

19
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gate2014ec1
numericalanswers
linearalgebra
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0
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0
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25
GATE ECE 2013  Question: 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 $k \times k$ identity matrix. Using the above property, the determinant of the matrix given below ... $2$ $5$ $8$ $16$
asked
Mar 26, 2018
in
Linear Algebra
by
Milicevic3306
(
15.8k
points)

16
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gate2013ec
linearalgebra
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0
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0
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26
GATE ECE 2012  Question: 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
Linear Algebra
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Milicevic3306
(
15.8k
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15
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gate2012ec
linearalgebra
matrices
0
votes
0
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27
GATE ECE 2018  Question: 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
Linear Algebra
by
gatecse
(
1.5k
points)

26
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gate2018ec
numericalanswers
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0
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0
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28
GATE ECE 2018  Question: 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 nonsingular (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
Linear Algebra
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gatecse
(
1.5k
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35
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gate2018ec
linearalgebra
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0
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0
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29
GATE ECE 2017 Set 2  Question: 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)

45
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gate2017ec2
linearalgebra
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rankofmatrix
numericalanswers
0
votes
0
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30
GATE ECE 2017 Set 1  Question: 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
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admin
(
2.8k
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135
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gate2017ec1
linearalgebra
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0
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31
GATE ECE 2017 Set 1  Question: 2
The rank of the matrix $\textbf{M} = \begin{bmatrix} 5&10&10 \\ 1 &0 &2 \\ 3&6&6 \end{bmatrix}$ is $0$ $1$ $2$ $3$
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Nov 17, 2017
in
Linear Algebra
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admin
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2.8k
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372
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