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Recent questions and answers in Linear Algebra
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UGCNETDec2004II: 1
AVA=A is called : Identity law De Morgan s law Idempotent law Complement law
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jothee
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2
NIELIT ScientistB Dec 2017_14
The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is _____. (A) 269 (B) 270 (C) 271 (D) 272
answered
Mar 19
in
Linear Algebra
by
topper98
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1
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ISI2015MMA42
Let $\lambda_1, \lambda_2, \lambda_3$ denote the eigenvalues of the matrix $A \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 &  \sin t & \cos t \end{pmatrix}.$ If $\lambda_1+\lambda_2+\lambda_3 = \sqrt{2}+1$ ... $\{  \frac{\pi}{4}, \frac{\pi}{4} \}$ $\{  \frac{\pi}{3}, \frac{\pi}{3} \}$
answered
Mar 14
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Linear Algebra
by
haralk10
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isi2015mma
linearalgebra
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1
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4
TIFR2020A2
Let $M$ be a real $n\times n$ matrix such that for every nonzero vector $x\in \mathbb{R}^{n},$ we have $x^{T}M x> 0.$ Then Such an $M$ cannot exist Such $Ms$ exist and their rank is always $n$ Such $Ms$ exist, but their eigenvalues are always real No eigenvalue of any such $M$ can be real None of the above
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Mar 8
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Linear Algebra
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ankitgupta.1729
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tifr2020
engineeringmathematics
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rankofmatrix
eigenvalue
+1
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1
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5
GF MT 4
Is the answer and explaination given correct ?
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Mar 4
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Linear Algebra
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smsubham
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gateforumtestseries
linearalgebra
rankofmatrix
+4
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3
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6
ISI2015MMA39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are $1,1,4$ $1,4,4$ $0,1,4$ $0,4,4$
answered
Mar 3
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Linear Algebra
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smsubham
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isi2015mma
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+3
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3
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7
GATE2020CS27
Let $A$ and $B$ be two $n \times n$ matrices over real numbers. Let rank($M$) and $\text{det}(M)$ denote the rank and determinant of a matrix $M$, respectively. Consider the following statements. $\text{rank}(AB) = \text{rank }(A) \text{rank }(B)$ ... Which of the above statements are TRUE? I and II only I and IV only II and III only III and IV only
answered
Feb 26
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Linear Algebra
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immanujs
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gate2020cs
discretemathematics
engineeringmathematics
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+2
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5
answers
8
GATE2020CS18
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
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Feb 26
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Linear Algebra
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immanujs
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9
Introduction to Linear Algebra 4th edition Problem Set 1.1
How many corner does a cube have in 4 dimensions? How many 3D faces? Now by observation we can tell that, an ndimensional cube has $2^n$ corners. 1D cube which is a line have $2^1$ corners 2D cube which is a square have $2^2$ ... . but this is the question i'm not able to answer. How every Ncube have $2n$ cubes of dimension (N1)?
asked
Feb 26
in
Linear Algebra
by
Mk Utkarsh
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1
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10
TIFR2020A12
The hour needle of a clock is malfunctioning and travels in the anticlockwise direction, i.e., opposite to the usual direction, at the same speed it would have if it was working correctly. The minute needle is working correctly. Suppose the the two needles show the correct time at ... $\dfrac{10}{11}$ hour $\dfrac{11}{12}$ hour $\dfrac{12}{13}$ hour $\dfrac{19}{22}$ hour One hour
answered
Feb 19
in
Linear Algebra
by
ankitgupta.1729
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18.1k
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45
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tifr2020
0
votes
1
answer
11
TIFR2020A5
Let $A$ be am $n\times n$ invertible matrix with real entries whose column sums are all equal to $1.$ Consider the following statements: Every column in the matrix $A^{2}$ sums to $2.$ Every column in the matrix $A^{3}$ sums to $3.$ Every column in the matrix $A^{1}$ ... $(1)\:\text{or}\:(2)$ all the $3$ statements $(1),(2),$ and $(3)$ are correct
answered
Feb 11
in
Linear Algebra
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Lakshman Patel RJIT
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tifr2020
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0
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12
TIFR2020A3
Let $d\geq 4$ and fix $w\in \mathbb{R}.$ Let $S = \{a = (a_{0},a_{1},\dots ,a_{d})\in \mathbb{R}^{d+1}\mid f_{a}(w) = 0\: \text{and}\: f'_{a}(w) = 0\},$ where the polynomial function $f_{a}(x)$ ... is a $d$dimensional vector subspace of $\mathbb{R}^{d+1}$ $S$ is a $(d1)$dimensional vector subspace of $\mathbb{R}^{d+1}$ None of the other options
asked
Feb 10
in
Linear Algebra
by
Lakshman Patel RJIT
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30
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tifr2020
engineeringmathematics
linearalgebra
vectorspace
+16
votes
6
answers
13
GATE2006IT26
What are the eigenvalues of the matrix $P$ given below $P= \begin{pmatrix} a &1 &0 \\ 1& a& 1\\ 0&1 &a \end{pmatrix}$ $a, a √2, a + √2$ $a, a, a$ $0, a, 2a$ $a, 2a, 2a$
answered
Feb 6
in
Linear Algebra
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gate2006it
linearalgebra
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0
votes
2
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14
A is a 4square matrix and A 5 = 0. Then
A is a 4square matrix and A_5 (a raised to the power of 5) = 0. Then A_4 = a) I b) I c) 0 d) A
answered
Jan 31
in
Linear Algebra
by
smsubham
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16k
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865
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matrices
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+34
votes
5
answers
15
GATE201826
Consider a matrix P whose only eigenvectors are the multiples of $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$. Consider the following statements. P does not have an inverse P has a repeated eigenvalue P cannot be diagonalized Which one of the following options ... and III are necessarily true Only II is necessarily true Only I and II are necessarily true Only II and III are necessarily true
answered
Jan 29
in
Linear Algebra
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JashanArora
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gate2018
linearalgebra
matrices
eigenvalue
normal
+35
votes
4
answers
16
GATE2016204
Consider the system, each consisting of $m$ linear equations in $n$ variables. If $m < n$, then all such systems have a solution. If $m > n$, then none of these systems has a solution. If $m = n$, then there exists a system which has a solution. Which one of the ... is CORRECT? $I, II$ and $III$ are true. Only $II$ and $III$ are true. Only $III$ is true. None of them is true.
answered
Jan 27
in
Linear Algebra
by
JashanArora
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4.9k
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gate20162
linearalgebra
systemofequations
normal
+18
votes
2
answers
17
TIFR2012B12
Let $A$ be a matrix such that $A^{k}=0$. What is the inverse of $I  A$? $0$ $I$ $A$ $1 + A + A^{2} + ...+ A^{k  1}$ Inverse is not guaranteed to exist.
answered
Jan 26
in
Linear Algebra
by
blackcloud
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675
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976
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tifr2012
linearalgebra
matrices
+36
votes
7
answers
18
GATE2007IT2
Let $A$ be the matrix $\begin{bmatrix}3 &1 \\ 1&2\end{bmatrix}$. What is the maximum value of $x^TAx$ where the maximum is taken over all $x$ that are the unit eigenvectors of $A?$ $5$ $\frac{(5 + √5)}{2}$ $3$ $\frac{(5  √5)}{2}$
answered
Jan 26
in
Linear Algebra
by
blackcloud
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675
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4.1k
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gate2007it
linearalgebra
eigenvalue
normal
+2
votes
2
answers
19
matrix groups
Which of the following is true? Every lower triangular matrix is group under multiplication operation where all elements of diagonal are non zero numbers. Every diagonal matrix is group under multiplication operation, where all elements of diagonal are non zero numbers. Every ... addition operation where all elements are real numbers. Both (a) and b) isn't a b c all are correct?
answered
Jan 4
in
Linear Algebra
by
Sahin
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361
points)

408
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grouptheory
+16
votes
3
answers
20
Mathematics: GATE 2013 ECA27
Let A be an mxn matrix and B an nxm matrix. It is given that determinant ( Im + AB ) = determinant ( In + BA ) , where Ik is the k k identity matrix. Using the above property, the determinant of the matrix given below is ... A) 2 B) 5 C) 8 D) 16
answered
Jan 4
in
Linear Algebra
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Satbir
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gate2013ec
linearalgebra
engineeringmathematics
normal
determinant
+10
votes
4
answers
21
GATE201944
Consider the following matrix: $R = \begin{bmatrix} 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{bmatrix}$ The absolute value of the product of Eigen values of $R$ is _______
answered
Jan 2
in
Linear Algebra
by
ankitgupta.1729
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4.4k
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gate2019
numericalanswers
engineeringmathematics
linearalgebra
eigenvalue
+19
votes
5
answers
22
GATE200426
The number of different $n \times n $ symmetric matrices with each element being either 0 or 1 is: (Note: $\text{power} \left(2, X\right)$ is same as $2^X$) $\text{power} \left(2, n\right)$ $\text{power} \left(2, n^2\right)$ $\text{power} \left(2,\frac{ \left(n^2+ n \right) }{2}\right)$ $\text{power} \left(2, \frac{\left(n^2  n\right)}{2}\right)$
answered
Jan 1
in
Linear Algebra
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JashanArora
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4k
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gate2004
linearalgebra
normal
matrices
0
votes
1
answer
23
ISI2017DCG25
If $f(x) = \begin{vmatrix} 2 \cos ^2 x & \sin 2x & – \sin x \\ \sin 2x & 2 \sin ^2 x & \cos x \\ \sin x & – \cos x & 0 \end{vmatrix},$ then $\int_0^{\frac{\pi}{2}} [ f(x) + f’(x)] dx$ is $\pi$ $\frac{\pi}{2}$ $0$ $1$
answered
Dec 30, 2019
in
Linear Algebra
by
ajaysoni1924
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11.1k
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56
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isi2017dcg
linearalgebra
determinant
definiteintegrals
nongate
+1
vote
1
answer
24
ISI2015MMA37
Let $a$ be a nonzero real number. Define $f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$ for $x \in \mathbb{R}$. Then, the number of distinct real roots of $f(x) =0$ is $1$ $2$ $3$ $4$
answered
Dec 30, 2019
in
Linear Algebra
by
ajaysoni1924
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11.1k
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83
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isi2015mma
linearalgebra
determinant
functions
+36
votes
5
answers
25
GATE201713
Let $c_{1}.....c_{n}$ be scalars, not all zero, such that $\sum_{i=1}^{n}c_{i}a_{i}$ = 0 where $a_{i}$ are column vectors in $R^{n}$. Consider the set of linear equations $Ax = b$ ... of equations has a unique solution at $x=J_{n}$ where $J_{n}$ denotes a $n$dimensional vector of all 1. no solution infinitely many solutions finitely many solutions
answered
Dec 25, 2019
in
Linear Algebra
by
mehul vaidya
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6.9k
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gate20171
linearalgebra
systemofequations
normal
+1
vote
1
answer
26
ISI2014DCG38
Suppose that $A$ is a $3 \times 3$ real matrix such that for each $u=(u_1, u_2, u_3)’ \in \mathbb{R}^3, \: u’Au=0$ where $u’$ stands for the transpose of $u$. Then which one of the following is true? $A’=A$ $A’=A$ $AA’=I$ None of these
answered
Dec 16, 2019
in
Linear Algebra
by
aiyyar.aarushi
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441
points)

92
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isi2014dcg
linearalgebra
matrices
0
votes
1
answer
27
characteristics polynomial
A is 5×5 matrix, all of whose entries are 1, then (a) A is not diagonalizable (b) A is idempotent (c) A is nilpotent (d) The minimal polynomial and the characteristics polynomial of A are not equal
answered
Dec 13, 2019
in
Linear Algebra
by
facil
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11
points)

102
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+6
votes
4
answers
28
ISRO200834
If a square matrix A satisfies $A^TA=I$, then the matrix $A$ is Idempotent Symmetric Orthogonal Hermitian
answered
Dec 3, 2019
in
Linear Algebra
by
JashanArora
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8.3k
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isro2008
linearalgebra
matrices
+8
votes
5
answers
29
GATE20199
Let $X$ be a square matrix. Consider the following two statements on $X$. $X$ is invertible Determinant of $X$ is nonzero Which one of the following is TRUE? I implies II; II does not imply I II implies I; I does not imply II I does not imply II; II does not imply I I and II are equivalent statements
answered
Nov 30, 2019
in
Linear Algebra
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JashanArora
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gate2019
engineeringmathematics
linearalgebra
determinant
+8
votes
2
answers
30
GATE19952.13
A unit vector perpendicular to both the vectors $a=2i3j+k$ and $b=i+j2k$ is: $\frac{1}{\sqrt{3}} (i+j+k)$ $\frac{1}{3} (i+jk)$ $\frac{1}{3} (ijk)$ $\frac{1}{\sqrt{3}} (i+jk)$
answered
Nov 28, 2019
in
Linear Algebra
by
Satbir
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1.2k
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gate1995
linearalgebra
normal
vectorspace
0
votes
1
answer
31
ISI2015MMA44
Let $P_1$, $P_2$ and $P_3$ denote, respectively, the planes defined by $\begin{array} {} a_1x +b_1y+c_1z=\alpha _1 \\ a_2x +b_2y+c_2z=\alpha _2 \\ a_3x +b_3y+c_3z=\alpha _3 \end{array}$ It is given that $P_1$, $P_2$ and $P_3$ ... then the planes do not have any common point of intersection intersect at a unique point intersect along a straight line intersect along a plane
answered
Nov 25, 2019
in
Linear Algebra
by
Lakshman Patel RJIT
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61.3k
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40
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isi2015mma
linearalgebra
systemofequations
0
votes
1
answer
32
ISI2014DCG64
The value of $\lambda$ such that the system of equation $\begin{array}{} 2x & – & y & + & 2z & = & 2 \\ x & – & 2y & + & z & = & 4 \\ x & + & y & + & \lambda z & = & 4 \end{array}$ has no solution is $3$ $1$ $0$ $3$
answered
Nov 24, 2019
in
Linear Algebra
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Lakshman Patel RJIT
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isi2014dcg
linearalgebra
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+1
vote
1
answer
33
ISI2016DCG31
Let $A$ be an $n\times n$ matrix such that $\mid\: A^{2}\mid=1.\:\: \mid A\:\mid$ stands for determinant of matrix $A.$ Then $\mid\:(A)\mid=1$ $\mid\:(A)\mid=0\:\text{or}\:1$ $\mid\:(A)\mid=1,0\:\text{or}\:1$ $\mid\:(A)\mid=1\:\text{or}\:1$
answered
Nov 18, 2019
in
Linear Algebra
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Lakshman Patel RJIT
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38
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isi2016dcg
linearalgebra
matrices
determinant
+4
votes
3
answers
34
ISRO200709
Eigen vectors of $\begin{bmatrix} 1 && \cos \theta \\ \cos \theta && 1 \end{bmatrix}$ are $\begin{bmatrix} a^n && 1 \\ 0 && a^n \end{bmatrix}$ $\begin{bmatrix} a^n && n \\ 0 && a^n \end{bmatrix}$ ... $\begin{bmatrix} a^n && na^{n1} \\ n && a^n \end{bmatrix}$
answered
Nov 16, 2019
in
Linear Algebra
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Viplove04
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isro2007
linearalgebra
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+5
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5
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35
Eigen Value
An orthogonal matrix A has eigen values 1, 2 and 4. What is the trace of the matrix
answered
Nov 12, 2019
in
Linear Algebra
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JashanArora
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matrices
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3
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36
ISI2014DCG8
If $M$ is a $3 \times 3$ matrix such that $\begin{bmatrix} 0 & 1 & 2 \end{bmatrix}M=\begin{bmatrix}1 & 0 & 0 \end{bmatrix}$ and $\begin{bmatrix}3 & 4 & 5 \end{bmatrix} M = \begin{bmatrix}0 & 1 & 0 \end{bmatrix}$ ... $\begin{bmatrix} 1 & 2 & 0 \end{bmatrix}$ $\begin{bmatrix} 9 & 10 & 8 \end{bmatrix}$
answered
Nov 11, 2019
in
Linear Algebra
by
techbd123
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isi2014dcg
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2
answers
37
TIFR2018A14
Let $A$ be an $n\times n$ invertible matrix with real entries whose row sums are all equal to $c$. Consider the following statements: Every row in the matrix $2A$ sums to $2c$. Every row in the matrix $A^{2}$ sums to $c^{2}$. Every row in the matrix $A^{1}$ ... $(1)$ and $(2)$ are correct but not necessarily statement $(3)$ all the three statements $(1), (2),$ and $(3)$ are correct
answered
Nov 9, 2019
in
Linear Algebra
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rohith1001
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tifr2018
matrices
linearalgebra
+16
votes
2
answers
38
TIFR2013B3
How many $4 \times 4$ matrices with entries from ${0, 1}$ have odd determinant? Hint: Use modulo $2$ arithmetic. $20160$ $32767$ $49152$ $57343$ $65520$
answered
Nov 9, 2019
in
Linear Algebra
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rohith1001
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tifr2013
linearalgebra
matrices
+21
votes
4
answers
39
GATE201211
Let A be the $ 2 × 2 $ matrix with elements $a_{11} = a_{12} = a_{21} = +1 $ and $ a_{22} = −1 $ . Then the eigenvalues of the matrix $A^{19}$ are $1024$ and $−1024$ $1024\sqrt{2}$ and $−1024 \sqrt{2}$ $4 \sqrt{2}$ and $−4 \sqrt{2}$ $512 \sqrt{2}$ and $−512 \sqrt{2}$
answered
Oct 31, 2019
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Linear Algebra
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Praveenk99
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99
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gate2012
linearalgebra
eigenvalue
+22
votes
5
answers
40
GATE2015333
If the following system has nontrivial solution, $px + qy + rz = 0$ $qx + ry + pz = 0$ $rx + py + qz = 0$, then which one of the following options is TRUE? $p  q + r = 0 \text{ or } p = q = r$ $p + q  r = 0 \text{ or } p = q = r$ $p + q + r = 0 \text{ or } p = q = r$ $p  q + r = 0 \text{ or } p = q = r$
answered
Oct 30, 2019
in
Linear Algebra
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techbd123
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gate20153
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