Recent questions and answers in Linear Algebra - GATE Overflow

+2 votes

2 answers

1

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) | 385 views

+16 votes

3 answers

2

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 (23.8k points) | 1.3k views

+7 votes

4 answers

3

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 (17k points) | 3.4k views

+17 votes

5 answers

4

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 (6.2k points) | 3.8k views

0 votes

1 answer

5

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 (10.8k points) | 35 views

+1 vote

1 answer

6

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 (10.8k points) | 53 views

+34 votes

5 answers

7

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 (5.3k points) | 6.3k views

+1 vote

1 answer

8

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 (413 points) | 59 views

0 votes

1 answer

9

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) | 80 views

+5 votes

4 answers

10

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 (6.2k points) | 1.5k views

+6 votes

5 answers

11

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 (6.2k points) | 2.3k views

+7 votes

2 answers

12

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 (23.8k points) | 1.1k views

0 votes

1 answer

13

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 (58.7k points) | 27 views

+2 votes

2 answers

14

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 Nov 25, 2019 in Linear Algebra by Lakshman Patel RJIT Veteran (58.7k points) | 74 views

0 votes

1 answer

15

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 (58.7k points) | 59 views

+1 vote

1 answer

16

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 (58.7k points) | 22 views

+4 votes

3 answers

17

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

+4 votes

5 answers

18

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 (6.2k points) | 330 views

+1 vote

3 answers

19

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 (3.5k points) | 113 views

+4 votes

2 answers

20

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 (1.9k points) | 555 views

+15 votes

2 answers

21

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 (1.9k points) | 1.2k views

+20 votes

4 answers

22

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 (83 points) | 2.9k views

+21 votes

5 answers

23

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 (3.5k points) | 3.2k views

+27 votes

6 answers

24

GATE2017-2-52
If the characteristic polynomial of a 3 $\times$ 3 matrix $M$ over $\mathbb{R}$ (the set of real numbers) is $\lambda^3 – 4 \lambda^2 + a \lambda +30, \quad a \in \mathbb{R}$, and one eigenvalue of $M$ is 2, then the largest among the absolute values of the eigenvalues of $M$ is _______

answered Oct 28, 2019 in Linear Algebra by techbd123 Active (3.5k points) | 4.1k views

+19 votes

2 answers

25

GATE1997-4.2
Let $A=(a_{ij})$ be an $n$-rowed square matrix and $I_{12}$ be the matrix obtained by interchanging the first and second rows of the $n$-rowed Identify matrix. Then $AI_{12}$ is such that its first Row is the same as its second row Row is the same as the second row of $A$ Column is the same as the second column of $A$ Row is all zero

answered Oct 24, 2019 in Linear Algebra by Satbir Boss (23.8k points) | 1.1k views

+24 votes

2 answers

26

GATE2017-2-22
Let $P = \begin{bmatrix}1 & 1 & -1 \\2 & -3 & 4 \\3 & -2 & 3\end{bmatrix}$ and $Q = \begin{bmatrix}-1 & -2 &-1 \\6 & 12 & 6 \\5 & 10 & 5\end{bmatrix}$ be two matrices. Then the rank of $ P+Q$ is ___________ .

answered Oct 24, 2019 in Linear Algebra by Satbir Boss (23.8k points) | 3.1k views

+9 votes

3 answers

27

GATE2005-49
What are the eigenvalues of the following $2\times 2$ matrix? $\left( \begin{array}{cc} 2 & -1\\ -4 & 5\end{array}\right)$ $-1$ and $1$ $1$ and $6$ $2$ and $5$ $4$ and $-1$

answered Oct 23, 2019 in Linear Algebra by Satbir Boss (23.8k points) | 962 views

+1 vote

1 answer

28

answered Oct 22, 2019 in Linear Algebra by aditi19 Active (5.2k points) | 55 views

0 votes

1 answer

29

answered Oct 22, 2019 in Linear Algebra by aditi19 Active (5.2k points) | 54 views

+1 vote

1 answer

30

ISI2015-MMA-61
Let $ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix} \text{ and } B=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}.$ Then there exists a matrix $C$ ... no matrix $C$ such that $A=BC$ there exists a matrix $C$ such that $A=BC$, but $A \neq CB$ there is no matrix $C$ such that $A=CB$

answered Oct 19, 2019 in Linear Algebra by chirudeepnamini Active (4.3k points) | 37 views

0 votes

1 answer

31

ISI2015-MMA-43
The values of $\eta$ for which the following system of equations $\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & = & \eta ^2 \end{array}$ has a solution are $\eta = 1, -2$ $\eta = -1, -2$ $\eta = 3, -3$ $\eta = 1, 2$

answered Oct 19, 2019 in Linear Algebra by chirudeepnamini Active (4.3k points) | 27 views

+9 votes

5 answers

32

GATE2007-IT-80
Let $P_{1},P_{2},\ldots,P_{n}$ be $n$ points in the $xy-$plane such that no three of them are collinear. For every pair of points $P_{i}$ and $P_{j}$, let $L_{ij}$ be the line passing through them. Let $L_{ab}$ be the line with the ... largest or the smallest $y$-coordinate among all the points The difference between $x$-coordinates $P_{a}$ and $P_{b}$ is minimum None of the above

answered Oct 4, 2019 in Linear Algebra by sujeetkumar (15 points) | 1.2k views

+1 vote

1 answer

33

ISI2014-DCG-9
The values of $\eta$ for which the following system of equations $\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & = & \eta ^2 \end{array}$ has a solution are $\eta=1, -2$ $\eta=-1, -2$ $\eta=3, -3$ $\eta=1, 2$

answered Sep 29, 2019 in Linear Algebra by techbd123 Active (3.5k points) | 74 views

0 votes

1 answer

34

ISI2015-MMA-63
Let $\theta=2\pi/67$. Now consider the matrix $A = \begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix}$. Then the matrix $A^{2010}$ ... $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

answered Sep 25, 2019 in Linear Algebra by `JEET Boss (18.9k points) | 23 views

0 votes

1 answer

35

ISI2015-MMA-62
If the matrix $A = \begin{bmatrix} a & 1 \\ 2 & 3 \end{bmatrix}$ has $1$ as an eigenvalue, then $\textit{trace}(A)$ is $4$ $5$ $6$ $7$

answered Sep 25, 2019 in Linear Algebra by Verma Ashish Boss (12.9k points) | 47 views

+1 vote

1 answer

36

ISI2014-DCG-25
The determinant $\begin{vmatrix} b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y \end{vmatrix}$ equals $\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ ... $3\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ None of these

answered Sep 24, 2019 in Linear Algebra by `JEET Boss (18.9k points) | 60 views

0 votes

0 answers

37

ISI2014-DCG-70
For the matrices $A = \begin{pmatrix} a & a \\ 0 & a \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, $(B^{-1}AB)^3$ is equal to $\begin{pmatrix} a^3 & a^3 \\ 0 & a^3 \end{pmatrix}$ ... $\begin{pmatrix} a^3 & 0 \\ 3a^3 & a^3 \end{pmatrix}$ $\begin{pmatrix} a^3 & 0 \\ -3a^3 & a^3 \end{pmatrix}$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (430k points) | 35 views

0 votes

0 answers

38

ISI2015-MMA-38
A real $2 \times 2$ matrix $M$ such that $M^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1- \varepsilon \end{pmatrix}$ exists for all $\varepsilon > 0$ does not exist for any $\varepsilon > 0$ exists for some $\varepsilon > 0$ none of the above is true

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (430k points) | 25 views

0 votes

0 answers

39

ISI2015-MMA-40
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix $\textbf{A}$ ... $(\textbf{A})$ equals $1$ or $2$ $0$ $4$ $2$ or $3$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (430k points) | 35 views

0 votes

0 answers

40

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 $\lambda_1+\lambda_2+\lambda_3 = \sqrt{2}+1$ ... $\{ - \frac{\pi}{4}, \frac{\pi}{4} \}$ $\{ - \frac{\pi}{3}, \frac{\pi}{3} \}$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (430k points) | 39 views

To see more, click for all the questions in this category.

Recent questions and answers in Linear Algebra

50,737 questions

57,271 answers

198,140 comments

104,781 users