Recent questions and answers in Linear Algebra - GATE Overflow

+1 vote

0 answers

1

UGCNET-Dec2004-II: 1
AVA=A is called : Identity law De Morgan s law Idempotent law Complement law

asked 6 days ago in Linear Algebra by jothee Veteran (107k points) | 25 views

0 votes

3 answers

2

NIELIT Scientist-B 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 Junior (589 points) | 228 views

0 votes

1 answer

3

ISI2015-MMA-42
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 in Linear Algebra by haralk10 Active (1.2k points) | 76 views

0 votes

1 answer

4

TIFR2020-A-2
Let $M$ be a real $n\times n$ matrix such that for every non-zero 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

answered Mar 8 in Linear Algebra by ankitgupta.1729 Boss (18.1k points) | 46 views

+1 vote

1 answer

5

GF MT 4
Is the answer and explaination given correct ?

answered Mar 4 in Linear Algebra by smsubham Boss (16k points) | 161 views

+4 votes

3 answers

6

ISI2015-MMA-39
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 in Linear Algebra by smsubham Boss (16k points) | 139 views

+3 votes

3 answers

7

GATE2020-CS-27
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 in Linear Algebra by immanujs Active (3.3k points) | 829 views

+2 votes

5 answers

8

GATE2020-CS-18
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.

answered Feb 26 in Linear Algebra by immanujs Active (3.3k points) | 1k views

0 votes

0 answers

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 n-dimensional 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 N-cube have $|2n|$ cubes of dimension (N-1)?

asked Feb 26 in Linear Algebra by Mk Utkarsh Boss (36.9k points) | 96 views

0 votes

1 answer

10

TIFR2020-A-12
The hour needle of a clock is malfunctioning and travels in the anti-clockwise di-rection, 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 Boss (18.1k points) | 45 views

0 votes

1 answer

11

TIFR2020-A-5
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 by Lakshman Patel RJIT Veteran (61.3k points) | 64 views

0 votes

0 answers

12

TIFR2020-A-3
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 $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1}$ None of the other options

asked Feb 10 in Linear Algebra by Lakshman Patel RJIT Veteran (61.3k points) | 30 views

+16 votes

6 answers

13

GATE2006-IT-26
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 by JashanArora Loyal (8.3k points) | 1.8k views

0 votes

2 answers

14

A is a 4-square matrix and A 5 = 0. Then
A is a 4-square 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 Boss (16k points) | 865 views

+34 votes

5 answers

15

GATE2018-26
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 by JashanArora Loyal (8.3k points) | 7.7k views

+35 votes

4 answers

16

GATE2016-2-04
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 Loyal (8.3k points) | 4.9k views

+18 votes

2 answers

17

TIFR2012-B-12
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 Junior (675 points) | 976 views

+36 votes

7 answers

18

GATE2007-IT-2
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 Junior (675 points) | 4.1k views

+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 (361 points) | 408 views

+16 votes

3 answers

20

Mathematics: GATE 2013 EC-A-27
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 by Satbir Boss (25.4k points) | 1.4k views

+10 votes

4 answers

21

GATE2019-44
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 Boss (18.1k points) | 4.4k views

+19 votes

5 answers

22

GATE2004-26
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 by JashanArora Loyal (8.3k points) | 4k views

0 votes

1 answer

23

ISI2017-DCG-25
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 Boss (11.1k points) | 56 views

+1 vote

1 answer

24

ISI2015-MMA-37
Let $a$ be a non-zero 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 Boss (11.1k points) | 83 views

+36 votes

5 answers

25

GATE2017-1-3
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 Loyal (5.4k points) | 6.9k views

+1 vote

1 answer

26

ISI2014-DCG-38
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 (441 points) | 92 views

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 (11 points) | 102 views

+6 votes

4 answers

28

ISRO2008-34
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 Loyal (8.3k points) | 1.5k views

+8 votes

5 answers

29

GATE2019-9
Let $X$ be a square matrix. Consider the following two statements on $X$. $X$ is invertible Determinant of $X$ is non-zero 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 by JashanArora Loyal (8.3k points) | 2.7k views

+8 votes

2 answers

30

GATE1995-2.13
A unit vector perpendicular to both the vectors $a=2i-3j+k$ and $b=i+j-2k$ is: $\frac{1}{\sqrt{3}} (i+j+k)$ $\frac{1}{3} (i+j-k)$ $\frac{1}{3} (i-j-k)$ $\frac{1}{\sqrt{3}} (i+j-k)$

answered Nov 28, 2019 in Linear Algebra by Satbir Boss (25.4k points) | 1.2k views

0 votes

1 answer

31

ISI2015-MMA-44
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 Veteran (61.3k points) | 40 views

0 votes

1 answer

32

ISI2014-DCG-64
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 by Lakshman Patel RJIT Veteran (61.3k points) | 82 views

+1 vote

1 answer

33

ISI2016-DCG-31
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 by Lakshman Patel RJIT Veteran (61.3k points) | 38 views

+4 votes

3 answers

34

ISRO2007-09
Eigen vectors of $\begin{bmatrix} 1 && \cos \theta \\ \cos \theta && 1 \end{bmatrix}$ are $\begin{bmatrix} a^n && 1 \\ 0 && a^n \end{bmatrix}$ $\begin{bmatrix} a^n && n \\ 0 && a^n \end{bmatrix}$ ... $\begin{bmatrix} a^n && na^{n-1} \\ -n && a^n \end{bmatrix}$

answered Nov 16, 2019 in Linear Algebra by Viplove04 (39 points) | 2.5k views

+5 votes

5 answers

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 by JashanArora Loyal (8.3k points) | 392 views

+1 vote

3 answers

36

ISI2014-DCG-8
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 Active (3.7k points) | 163 views

+5 votes

2 answers

37

TIFR2018-A-14
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 by rohith1001 Active (2.3k points) | 612 views

+16 votes

2 answers

38

TIFR2013-B-3
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 by rohith1001 Active (2.3k points) | 1.3k views

+21 votes

4 answers

39

GATE2012-11
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 in Linear Algebra by Praveenk99 (99 points) | 3.1k views

+22 votes

5 answers

40

GATE2015-3-33
If the following system has non-trivial 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 by techbd123 Active (3.7k points) | 3.4k views

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