Recent questions tagged matrices

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TIFR ECE 2015 | Question: 6
$\textbf{A}$ is an $n \times n$ square matrix of reals such that $\mathbf{A y}=\mathbf{A}^{T} \mathbf{y}$, for all real vectors $\mathbf{y}$. Which of the following can we conclude? $\mathbf{A}$ is invertible $\mathbf{A}^{T}=\mathbf{A}$ $\mathbf{A}^{2}=\mathbf{A}$ Only (i) Only (ii) Only (iii) Only (i) and (ii) None of the above

admin asked in Linear Algebra Dec 15, 2022

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TIFR ECE 2014 | Question: 5
The matrix \[ A=\left(\begin{array}{ccc} 1 & a_{1} & a_{1}^{2} \\ 1 & a_{2} & a_{2}^{2} \\ 1 & a_{3} & a_{3}^{2} \end{array}\right) \] is invertible when $a_{1}>a_{2}>a_{3}$ $a_{1}<a_{2}<a_{3}$ $a_{1}=3, a_{2}=2, a_{3}=4$ All of the above None of the above

admin asked in Linear Algebra Dec 14, 2022

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TIFR ECE 2013 | Question: 13
Let $A$ be a square matrix and $x$ be a vector whose dimensions match $A$. Let $B^{\dagger}$ be the conjugate transpose of $B$. Then which of the following is not true: $x^{\dagger} A^{2} x$ is always non-negative $x^{\dagger} A x$ ... $A=A^{\dagger}$ then $x^{\dagger} A y$ is complex for some vector $y$ with same dimensions as $x$

admin asked in Linear Algebra Dec 12, 2022

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TIFR ECE 2012 | Question: 18
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the right side. If the transformation is of the form \[ \left[\begin{array}{l} u \\ v \end{array}\right]=\mathbf{A}\ ... \] $A_{1}$ only $A_{2}$ only $A_{1}$ or $A_{2}$ $A_{1}$ or $A_{3}$ $A_{2}$ or $A_{3}$

admin asked in Linear Algebra Dec 8, 2022

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TIFR ECE 2012 | Question: 20
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then $B=e^{2} A / 2$ $B=\left(\begin{array}{cc}1+e & e \\e & 1+e\end{array}\right)$ ... $B=\left(\begin{array}{cc}1+e^{2} & e^{2} \\e^{2} & 1+e^{2}\end{array}\right)$ None of the above

admin asked in Linear Algebra Dec 8, 2022

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TIFR ECE 2010 | Question: 10
$\text{H}$ is a circulant matrix (row $n$ is obtained by circularly shifting row $1$ to the right by $n$ positions) and $\text{F}$ is the $\text{DFT}$ matrix. Which of the following is true? $F H F^{H}$ is circulant, where $F^{H}$ is the inverse $\text{DFT}$ matrix. $F H F^{H}$ is tridiagonal $F H F^{H}$ is diagonal $F H F^{H}$ has real entries None of the above

admin asked in Linear Algebra Dec 1, 2022

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7

TIFR ECE 2010 | Question: 11
Consider \[ \text{F}=\frac{1}{2}\left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \\ 1 & -1 & 1 & -1 \end{array}\right], \quad x=\left[\begin{array}{l} 2.1 \\ 1.2 \\ ... 2 \\ -1 \end{array}\right] \] The inner product between $\text{F}x$ and $\text{F}y$ is $0$ $1$ $-1$ $-1.2$ None of the above

admin asked in Linear Algebra Dec 1, 2022

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TIFR ECE 2022 | Question: 12
An $n \times n$ matrix $\mathbf{P}$ is called a Permutation Matrix if each of its $n$ columns and $n$ rows contain exactly one $1$ and $n-1 \; 0$ 's. Consider the following statements: $\operatorname{det}(\mathbf{P})$ is either $+1$ ... $1,3$ are correct Only statements $2, 3$ are correct All statements $1, 2,$ and $3$ are correct

admin asked in Linear Algebra Nov 30, 2022

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TIFR ECE 2020 | Question: 9
Let $A$ be an $n \times n$ matrix with the the property that $A^{m}=0$ for some $m \in \mathbb{N}$. Consider the following statements: At least one entry of $A$ is zero All eigenvalues of $A$ are zero All diagonal entries of $A$ are zero ... $2$ alone is correct Only statement $3$ is correct Only statements $1$ and $2$ are correct Only statements $2$ and $3$ are correct

admin asked in Linear Algebra Nov 30, 2022

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TIFR ECE 2020 | Question: 14
Two matrices $A$ and $B$ are called similar if there exists an invertible matrix $X$ such that $A=X^{-1} B X$. Let $A$ and $B$ be two similar matrices. Consider the following statements: $\operatorname{det}(x I-A)=\operatorname{det}(x I-B)$ ... statement $2$ is correct Only statements $1$ and $2$ are correct All Statements $1, 2$ and $3$ are correct None of the above

admin asked in Linear Algebra Nov 30, 2022

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11

TIFR ECE 2019 | Question: 2
Let $A$ and $B$ be two square matrices that have full rank. Let $\lambda_{A}$ be an eignevalue of $A$ and $\lambda_{B}$ an eigenvalue of $B$. Which of the following is always $\text{TRUE}?$ $A B$ has full rank $A-B$ ... an eigenvalue of $A B$ $A+B$ has full rank At least one of $\lambda_{A}$ or $\lambda_{B}$ is an eigenvalue of $A B$

admin asked in Linear Algebra Nov 30, 2022

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TIFR ECE 2018 | Question: 8
Let $A$ be an $n \times n$ real matrix for which two distinct non-zero $n$-dimensional real column vectors $v_{1}, v_{2}$ satisfy the relation $A v_{1}=A v_{2}$. Consider the following statements. At least one eigenvalue of $A$ is zero. $A$ ... $\text{(i)}$ Only $\text{(ii)}$ Only $\text{(iii)}$ Only $\text{(iv)}$ All of $\text{(i) - (iv)}$

admin asked in Linear Algebra Nov 29, 2022

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TIFR ECE 2017 | Question: 13
Let $A$ be an $n \times n$ matrix. Consider the following statements. $A$ can have full-rank even if there exists two vectors $v_{1} \neq v_{2}$ such that $A v_{1}=A v_{2}$. $A$ can be similar to the identity matrix, when $A$ is not the identity matrix. Recall that ... $\text{(ii)}$ Only $\text{(iii)}$ $\text{(i), (ii),}$ and $\text{(iii)}$ None of the above

admin asked in Linear Algebra Nov 29, 2022

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14

TIFR ECE 2016 | Question: 13
Suppose $m$ and $n$ are positive integers, $m \neq n$, and $A$ is an $m \times n$ matrix with real entires. Consider the following statements. $\operatorname{rank}\left(A A^{T}\right)=\operatorname{rank}\left(A^{T} A\right)$ ... Which of the above statements is true for all such $A?$ Only (i) Only (ii) Only (iii) (i) and (iii) None of them

admin asked in Linear Algebra Nov 29, 2022

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15

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 ____________.

Arjun asked in Linear Algebra Feb 12, 2019

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16

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 $+j\omega$ $-j\omega$ $+\omega$ $-\omega$

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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17

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 ________

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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18

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 $\mid a-b \mid$ is ________

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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19

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: $M^{4k+1}$ $M^{4k+2}$ $M^{4k+3}$ $M^{4k}$

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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20

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 _________

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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21

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 $\sec^{2}x$ $\cos 4x$ $1$ $0$

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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22

GATE ECE 2015 Set 2 | Question: 2
The value of $x$ for which all the eigen-values of the matrix given below are real is $\begin{bmatrix} 10&5+j &4 \\ x&20 &2 \\4 &2 &-10 \end{bmatrix}$ $5+j$ $5-j$ $1-5j$ $1+5j$

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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23

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 $\sin(t)$ $1-e^{t}$ $1-\cos(t)$ $0$

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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24

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 _________.

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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25

GATE ECE 2015 Set 1 | Question: 43
Two sequences $\begin{bmatrix}a, & b, & c \end{bmatrix}$ and $\begin{bmatrix}A, & B, & C \end{bmatrix}$ ... $\begin{bmatrix}p, & q, & r \end{bmatrix} = \begin{bmatrix} c, & b, & a \end{bmatrix}$

Milicevic3306 asked in Linear Algebra Mar 28, 2018

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26

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 $\begin{bmatrix} t & 1 \\ 1 & 0 \end{bmatrix} \\$ ... $\begin{bmatrix} 0 & 1 \\ 1 & t \end{bmatrix} \\$ $\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}$

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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27

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

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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28

GATE ECE 2014 Set 3 | Question: 47
The state equation of a second-order linear system is given by $\dot{x}(t)=Ax(t), \:\:\:\:\:\:\:\:x(0)=x_{0}$ For $x_{0}= \begin{bmatrix} 1\\ -1 \end{bmatrix},$ $x(t)= \begin{bmatrix} e^{-t}\\ -e^{-t} \end{bmatrix},$ ... $\begin{bmatrix} 5e^{-t}-3e^{-2t}\\ -5e^{-t}+6e^{-2t} \end{bmatrix}$

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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29

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 ________

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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30

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

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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31

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 __________.

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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32

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? $(M^{T})^{T} = M$ $(cM)^{T} = c(M)^{T}$ $(M+N)^{T} = M^{T} + N^{T}$ $MN = NM$

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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33

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 ______.

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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34

GATE ECE 2014 Set 1 | Question: 29
Consider the matrix ... $\alpha$ is a non-negative real number. The value of $\alpha$ for which $\text{det(P)} = 0$ is _______.

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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35

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 ... $2$ $5$ $8$ $16$

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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36

GATE ECE 2013 | Question: 19
The minimum eigenvalue of the following matrix is $\begin{bmatrix} 3& 5& 2\\5 &12 &7 \\2 &7 & 5\end{bmatrix}$ $0$ $1$ $2$ $3$

Milicevic3306 asked in Linear Algebra Mar 26, 2018

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37

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 $15\:A+12\:I$ $19\:A+30\:I$ $17\:A+15\:I$ $17\:A+21\:I$

Milicevic3306 asked in Linear Algebra Mar 25, 2018

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38

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 non-singular (invertible). Which one among the following is TRUE? $S1$ implies $S2$ $S1$ implies $S3$ $S2$ implies $S1$ $S3$ implies $S2$

gatecse asked in Linear Algebra Feb 19, 2018

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39

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 ________.

admin asked in Linear Algebra Nov 23, 2017

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40

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 $0$ $1$ $2$ $3$

admin asked in Linear Algebra Nov 17, 2017

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