Recent questions and answers in Linear Algebra

+1 vote

1 answer

1

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 4 hours ago in Linear Algebra by Arjun Veteran (406k points) | 259 views

+37 votes

3 answers

2

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 2 days ago in Linear Algebra by Ashwani Kumar 2 Boss (14.6k points) | 2.9k views

+2 votes

2 answers

3

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 (13.2k points) | 252 views

0 votes

1 answer

4

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 Junior (953 points) | 55 views

+17 votes

4 answers

5

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 (13.2k points) | 2.9k views

0 votes

1 answer

6

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 (111k points) | 54 views

+1 vote

4 answers

7

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 (349 points) | 88 views

+21 votes

3 answers

8

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 (13.2k points) | 2.7k views

0 votes

2 answers

9

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 (12.9k points) | 58 views

+1 vote

1 answer

10

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 (467 points) | 78 views

0 votes

1 answer

11

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 (223 points) | 106 views

+17 votes

3 answers

12

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 (1.2k points) | 2.8k views

0 votes

0 answers

13

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 (3k points) | 48 views

+8 votes

1 answer

14

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) | 508 views

+17 votes

3 answers

15

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 (1.2k points) | 2.4k views

0 votes

2 answers

16

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 (33.9k points) | 73 views

+1 vote

1 answer

17

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 (11 points) | 54 views

0 votes

1 answer

18

#GATE 2014 IN

answered May 14 in Linear Algebra by Satbir Boss (12.9k points) | 28 views

0 votes

1 answer

19

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 May 12 in Linear Algebra by Sayan Bose Loyal (6.9k points) | 42 views

0 votes

2 answers

20

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 May 11 in Linear Algebra by srestha Veteran (111k points) | 65 views

0 votes

1 answer

21

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 Loyal (7.3k points) | 53 views

0 votes

2 answers

22

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) | 173 views

0 votes

1 answer

23

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) | 113 views

0 votes

2 answers

24

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) | 145 views

0 votes

1 answer

25

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 (1.2k points) | 113 views

+10 votes

4 answers

26

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 (1.2k points) | 638 views

+1 vote

0 answers

27

CSIR UGC NET

asked Apr 28 in Linear Algebra by Hirak Active (3k points) | 36 views

+1 vote

0 answers

28

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 (3k points) | 50 views

+1 vote

2 answers

29

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

answered Apr 27 in Linear Algebra by SuvasishDutta Junior (655 points) | 58 views

+3 votes

1 answer

30

Vani Institute Question Bank Pg-231 chapter 6
The Eigen values of $A=\begin{bmatrix} a& 1& 0\\1 &a &1 \\0 &1 &a \end{bmatrix}$ are______ $a,a,a$ $0,a,2a$ $-a,2a,2a$ $a,a+\sqrt{2},a-\sqrt{2}$

answered Apr 27 in Linear Algebra by SuvasishDutta Junior (655 points) | 85 views

+3 votes

4 answers

31

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 Apr 23 in Linear Algebra by gaurav1.yuva (427 points) | 1.7k views

+2 votes

1 answer

32

TIFR-2015-Maths-B-13
Let $X=\left\{(x, y) \in \mathbb{R}^{2}: 2x^{2}+3y^{2}= 1\right\}$. Endow $\mathbb{R}^{2}$ with the discrete topology, and $X$ with the subspace topology. Then. $X$ is a compact subset of $\mathbb{R}^{2}$ in this topology. $X$ is a connected subset of $\mathbb{R}^{2}$ in this topology. $X$ is an open subset of $\mathbb{R}^{2}$ in this topology. None of the above.

answered Apr 8 in Linear Algebra by Sushantkala786 (11 points) | 77 views

0 votes

4 answers

33

ISRO 2012- ECE Matrices
The Eigen values of matrix are: a) ± cos∝ b) ± sin ∝ c) tan ∝ & cot ∝ d) cos ∝ ± sin ∝

answered Mar 10 in Linear Algebra by Debdeep1998 Junior (849 points) | 101 views

0 votes

3 answers

34

ISRO2012-ECE: Engineering Mathematics
The system of equations x + y + z = 6, 2x + y + z = 7, x + 2 y + z = 8 has a. A unique solution b. No solution c. An infinite number of solutions d. None of these

answered Mar 10 in Linear Algebra by abhishekmehta4u Boss (33.9k points) | 64 views

+4 votes

4 answers

35

Eigen Value
An orthogonal matrix A has eigen values 1, 2 and 4. What is the trace of the matrix

answered Mar 10 in Linear Algebra by Verma Ashish Loyal (7.3k points) | 240 views

0 votes

1 answer

36

Ace Test Series: Linear Algebra - Eigen Values

answered Mar 9 in Linear Algebra by abhishekmehta4u Boss (33.9k points) | 155 views

0 votes

2 answers

37

ISRO 2014 -Trigonometry [EE]
If tan $\Theta$ = 8 / 15 and $\Theta$ is acute, then cosec $\Theta$ (A) 8 / 17 (B) 8 / 15 (C) 17 / 8 (D) 17 / 15

answered Mar 7 in Linear Algebra by abhishekmehta4u Boss (33.9k points) | 207 views

0 votes

0 answers

38

ISI-MMA-2015-44
Let $P_{1},P_{2},$ and $P_{3}$ denote, respectively, the planes defined by $a_{1}x + b_{1}y + c_{1}z = \alpha _{1}$ $a_{2}x + b_{2}y + c_{2}z = \alpha _{2}$ $a_{3}x + b_{3}y + c_{3}z = \alpha _{3}$ It is given ... then the planes (A) do not have any common point of intersection (B) intersect at a unique point (C) intersect along a straight line (D) intersect along a plane

asked Feb 22 in Linear Algebra by ankitgupta.1729 Boss (13.2k points) | 81 views

0 votes

1 answer

39

Linear Algebra-METest
If the determinant of the below matrix is 245, what is the value of X? $\begin{bmatrix} 0 & 4& 2 &1 \\ 3& -1 & 0 & 2\\ 5&2 & x& 4\\ 6& 1 & -1 & 0 \end{bmatrix}$ ... , because by keeping 6 in place of x I am getting determinant as 245(Result verified by computer program). Please let me know where I am going wrong?

answered Feb 19 in Linear Algebra by Meet2698 (195 points) | 140 views

+20 votes

4 answers

40

GATE2015-2-27
Perform the following operations on the matrix $\begin{bmatrix} 3 & 4 & 45 \\ 7 & 9 & 105 \\ 13 & 2 & 195 \end{bmatrix}$ Add the third row to the second row Subtract the third column from the first column. The determinant of the resultant matrix is _____.

answered Feb 18 in Linear Algebra by Ram Swaroop Active (2.9k points) | 1.8k views

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