Recent questions and answers in Linear Algebra

+1 vote

1 answer

1

Gatebook Test

answered 23 hours ago in Linear Algebra by aditi19 Active (4.9k points) | 43 views

0 votes

1 answer

2

Gatebook Test

answered 23 hours ago in Linear Algebra by aditi19 Active (4.9k points) | 44 views

+8 votes

5 answers

3

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 in Linear Algebra by sujeetkumar (11 points) | 1k views

+1 vote

3 answers

4

ISI2018-MMA-12
The rank of the matrix $\begin{bmatrix} 1 &2 &3 &4 \\ 5& 6 & 7 & 8 \\ 6 & 8 & 10 & 12 \\ 151 & 262 & 373 & 484 \end{bmatrix}$ $1$ $2$ $3$ $4$

answered Sep 23 in Linear Algebra by aditi19 Active (4.9k points) | 108 views

0 votes

2 answers

5

ISI2018-MMA-13
If $A =\begin{bmatrix} 2 &i \\ i & 0 \end{bmatrix}$ , the trace of $A^{10}$ is $2$ $2(1+i)$ $0$ $2^{10}$

answered Sep 23 in Linear Algebra by aditi19 Active (4.9k points) | 73 views

+24 votes

5 answers

6

GATE2016-1-05
Two eigenvalues of a $3 \times 3$ real matrix $P$ are $(2+\sqrt {-1})$ and $3$. The determinant of $P$ is _______

answered Sep 19 in Linear Algebra by neeraj2681 (33 points) | 3.6k views

0 votes

2 answers

7

GATE-2007 EE
Let $x$ and $y$ be two vectors in a $3$ dimensional space and $<x,y>$ denote their dot product. Then the determinant $det\begin{bmatrix}<x,x> & <x,y>\\ <y,x> & <y,y>\end{bmatrix}$ is zero when $x$ and $y$ are linearly ... $x$ and $y$ are linearly independent is non-zero for all non-zero $x$ and $y$ is zero only when either $x$ or $y$ is zero

answered Sep 11 in Linear Algebra by Pratik Gawali Active (1k points) | 311 views

+10 votes

3 answers

8

GATE1994-1.9
The rank of matrix $\begin{bmatrix} 0 & 0 & -3 \\ 9 & 3 & 5 \\ 3 & 1 & 1 \end{bmatrix}$ is: $0$ $1$ $2$ $3$

answered Sep 7 in Linear Algebra by Prafful (13 points) | 966 views

+27 votes

5 answers

9

GATE2006-23
$F$ is an $n\times n$ real matrix. $b$ is an $n\times 1$ real vector. Suppose there are two $n\times 1$ vectors, $u$ and $v$ such that, $u ≠ v$ and $Fu = b, Fv = b$. Which one of the following statements is false? Determinant of $F$ is zero. There are an infinite number of solutions to $Fx = b$ There is an $x≠0$ such that $Fx = 0$ $F$ must have two identical rows

answered Aug 27 in Linear Algebra by JashanArora Junior (827 points) | 2.2k views

+15 votes

5 answers

10

GATE2005-IT-3
The determinant of the matrix given below is $\begin{bmatrix} 0 &1 &0 &2 \\ -1& 1& 1& 3\\ 0&0 &0 & 1\\ 1& -2& 0& 1 \end{bmatrix}$ $-1$ $0$ $1$ $2$

answered Aug 5 in Linear Algebra by Satbir Boss (19.5k points) | 1.9k views

+10 votes

3 answers

11

GATE2002-1.1
The rank of the matrix $\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$ is $4$ $2$ $1$ $0$

answered Jul 31 in Linear Algebra by thambu (11 points) | 866 views

0 votes

1 answer

12

MADE EASY
The Necessary condition to diagonalize a matrix is that A) ITS all eigen values should be distdist B) its eigen vectors should be indeindepent C) its eigen values should be real D) matrix is non singular

answered Jul 30 in Linear Algebra by Jyotish Ranjan (11 points) | 58 views

+46 votes

7 answers

13

GATE2014-2-47
The product of the non-zero eigenvalues of the matrix is ____ $\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}$

answered Jul 24 in Linear Algebra by srestha Veteran (116k points) | 9.6k views

+1 vote

1 answer

14

GATE1998-9
Derive the expressions for the number of operations required to solve a system of linear equations in $n$ unknowns using the Gaussian Elimination Method. Assume that one operation refers to a multiplication followed by an addition.

answered Jun 21 in Linear Algebra by suraj Loyal (6k points) | 304 views

+37 votes

3 answers

15

GATE2007-25
Let A be a $4 \times 4$ matrix with eigen values -5,-2,1,4. Which of the following is an eigen value of the matrix$\begin{bmatrix} A & I \\ I & A \end{bmatrix}$, where $I$ is the $4 \times 4$ identity matrix? $-5$ $-7$ $2$ $1$

answered Jun 15 in Linear Algebra by Ashwani Kumar 2 Boss (15.4k points) | 3.1k views

+2 votes

2 answers

16

GATE1988-16i
Assume that the matrix $A$ given below, has factorization of the form $LU=PA$, where $L$ is lower-triangular with all diagonal elements equal to 1, $U$ is upper-triangular, and $P$ is a permutation matrix. For $A = \begin{bmatrix} 2 & 5 & 9 \\ 4 & 6 & 5 \\ 8 & 2 & 3 \end{bmatrix}$ Compute $L, U,$ and $P$ using Gaussian elimination with partial pivoting.

answered Jun 8 in Linear Algebra by ankitgupta.1729 Boss (15.4k points) | 293 views

0 votes

1 answer

17

Linear Algebra (Self Doubt)
Let $A$ be a $n \times n$ square matrix whose all columns are independent. Is $Ax = b$ always solvable? Actually, I know that $Ax= b$ is solvable if $b$ is in the column space of $A$. However, I am not sure if it is solvable for all values of $b$.

answered Jun 7 in Linear Algebra by Sourajit25 Active (1.2k points) | 96 views

+17 votes

4 answers

18

GATE2007-27
Consider the set of (column) vectors defined by$X = \left \{x \in R^3 \mid x_1 + x_2 + x_3 = 0, \text{ where } x^T = \left[x_1,x_2,x_3\right]^T\right \}$ ... a linearly independent set, but it does not span $X$ and therefore is not a basis of $X$. $X$ is not a subspace of $R^3$. None of the above

answered Jun 6 in Linear Algebra by ankitgupta.1729 Boss (15.4k points) | 3.2k views

0 votes

1 answer

19

GATE 2019:EC
The value of integral $\int_{0}^{\pi }\int_{y}^{\pi }\frac{\sin x}{x}dxdy$ is equal to_________

answered Jun 3 in Linear Algebra by srestha Veteran (116k points) | 74 views

+2 votes

4 answers

20

GATE2017 EC
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 ________. Ans 5?

answered Jun 3 in Linear Algebra by Debargha Bhattacharj Junior (653 points) | 198 views

+20 votes

3 answers

21

GATE2015-1-1‏8
In the LU decomposition of the matrix $\begin{bmatrix}2 & 2 \\ 4 & 9\end{bmatrix}$, if the diagonal elements of $U$ are both $1$, then the lower diagonal entry $l_{22}$ of $L$ is_________________.

answered Jun 2 in Linear Algebra by ankitgupta.1729 Boss (15.4k points) | 2.8k views

0 votes

2 answers

22

GATE 2017:EC
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 $A$ has only one real eigen value. Then the real eigen value of $A$ is ________ [closed]

answered Jun 2 in Linear Algebra by Satbir Boss (19.5k points) | 152 views

+1 vote

1 answer

23

TIFR-2014-Maths-A-11
Let $A$ be an $n \times n$ matrix with real entries such that $A^{k}=0$ (0-matrix), for some $k \in \mathbb{N}$. Then $A$ has to be the $0$ matrix Trace$(A)$ could be non-zero $A$ is diagonalizable $0$ is the only eigenvalue of $A$.

answered May 31 in Linear Algebra by Yash4444 Junior (609 points) | 82 views

0 votes

1 answer

24

Self-Doubt: Diagonalizable Matrix
$1)$ How to find a matrix is diagonalizable or not? Suppose a matrix is $A=\begin{bmatrix} \cos \Theta &\sin \Theta \\ \sin\Theta & -\cos\Theta \end{bmatrix}$ Is it diagonalizable? $2)$ What is it’s eigen spaces?

answered May 30 in Linear Algebra by lolster (233 points) | 136 views

+16 votes

3 answers

25

GATE2003-41
Consider the following system of linear equations ... are linearly dependent. For how many values of $\alpha$, does this system of equations have infinitely many solutions? $0$ $1$ $2$ $3$

answered May 29 in Linear Algebra by MRINMOY_HALDER Active (2.6k points) | 3k views

0 votes

0 answers

26

GATE MOCK 2018
An orthogonal matrix A has eigen values 1, 2 and 4, then trace of the matrix $A^T$ is ___________ [closed]

asked May 28 in Linear Algebra by Hirak Active (3.5k points) | 69 views

+9 votes

1 answer

27

Mathematics GATE EE
The maximum value of a such that the matrix below has three linearly independent real eigen vectors is $\begin{pmatrix} -3& 0 &-2 \\ 1& -1 & 0\\ 0& a & 2 \end{pmatrix}$ (a) $\frac{2}{3\sqrt{3}}$ (b) $\frac{1}{3\sqrt{3}}$ (c) $\frac{1+2\sqrt{3}}{3\sqrt{3}}$ (d)$\frac{1+\sqrt{3}}{3\sqrt{3}}$

answered May 27 in Linear Algebra by Aishvarya Akshaya Vi (39 points) | 543 views

+18 votes

3 answers

28

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 May 27 in Linear Algebra by MRINMOY_HALDER Active (2.6k points) | 2.5k views

0 votes

2 answers

29

Engineering Maths
If A = $\begin{bmatrix} -1 & 1 & 0 \\ 0 & 2 &-2 \\ 0& 0 & 3 \end{bmatrix}$ then trace of the matrix 3A2 + adj A is ____

answered May 26 in Linear Algebra by abhishekmehta4u Boss (35.1k points) | 84 views

+1 vote

1 answer

30

ISI2018-PCB-A1
Consider a $n \times n$ matrix $A=I_n-\alpha\alpha^T$, where $I_n$ is the $n\times n$ identity matrix and $\alpha$ is an $n\times 1$ column vector such that $\alpha^T\alpha=1$.Show that $A^2=A$.

answered May 20 in Linear Algebra by Kaustubh Vande (21 points) | 64 views

0 votes

1 answer

31

#GATE 2014 IN

answered May 14 in Linear Algebra by Satbir Boss (19.5k points) | 31 views

0 votes

1 answer

32

ISI2018-MMA-14
Let $A$ be a $3× 3$ real matrix with all diagonal entries equal to $0$. If $1 + i$ is an eigenvalue of $A$, the determinant of $A$ equals $-4$ $-2$ $2$ $4$

answered May 11 in Linear Algebra by Verma Ashish Boss (10.7k points) | 72 views

0 votes

2 answers

33

ISI2019-MMA-13
Let $V$ be the vector space of all $4 \times 4$ matrices such that the sum of the elements in any row or any column is the same. Then the dimension of $V$ is $8$ $10$ $12$ $14$

answered May 9 in Linear Algebra by pratekag Active (2k points) | 187 views

+1 vote

1 answer

34

ISI2019-MMA-23
Let $A$ be $2 \times 2$ matrix with real entries. Now consider the function $f_A(x)$ = $Ax$ . If the image of every circle under $f_A$ is a circle of the same radius, then A must be an orthogonal matrix A must be a symmetric matrix A must be a skew-symmetric matrix None of the above must necessarily hold

answered May 7 in Linear Algebra by pratekag Active (2k points) | 125 views

0 votes

2 answers

35

ISI2019-MMA-15
The rank of the matrix $\begin{bmatrix} 0 &1 &t \\ 2& t & -1\\ 2& 2 & 0 \end{bmatrix}$ equals $3$ for any real number $t$ $2$ for any real number $t$ $2$ or $3$ depending on the value of $t$ $1,2$ or $3$ depending on the value of $t$

answered May 7 in Linear Algebra by Shikha Mallick Active (3.4k points) | 156 views

+1 vote

1 answer

36

ISI2019-MMA-14
If the system of equations $\begin{array} \\ax +y+z= 0 \\ x+by +z = 0 \\ x+y + cz = 0 \end{array}$ with $a,b,c \neq 1$ has a non trivial solutions, the value of $\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c}$ is $1$ $-1$ $3$ $-3$

answered May 7 in Linear Algebra by MRINMOY_HALDER Active (2.6k points) | 143 views

+10 votes

4 answers

37

GATE1996-10
Let $A = \begin{bmatrix} a_{11} && a_{12} \\ a_{21} && a_{22} \end{bmatrix} \text { and } B = \begin{bmatrix} b_{11} && b_{12} \\ b_{21} && b_{22} \end{bmatrix}$ be two matrices such that $AB=I$. Let $C = A \begin{bmatrix} 1 && 0 \\ 1 && 1 \end{bmatrix}$ and $CD =I$. Express the elements of $D$ in terms of the elements of $B$.

answered May 3 in Linear Algebra by MRINMOY_HALDER Active (2.6k points) | 685 views

+1 vote

0 answers

38

CSIR UGC NET

asked Apr 28 in Linear Algebra by Hirak Active (3.5k points) | 49 views

+1 vote

0 answers

39

Made Easy Engineering Maths book
The ans given is b, but i am not able to understande why. According to me the largest eigen value is 2, and therefore none of the option matches..!

asked Apr 27 in Linear Algebra by Hirak Active (3.5k points) | 81 views

+1 vote

2 answers

40

Virtual Gate Test Series: Linear Algebra - Rank Of The Matrix

answered Apr 27 in Linear Algebra by SuvasishDutta Active (1.3k points) | 65 views

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