Recent questions tagged matrices

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1

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

asked Sep 23 in Linear Algebra by Arjun Veteran (423k points) | 11 views

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2

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 $3$ $1$ $0$ $-3$

asked Sep 23 in Linear Algebra by Arjun Veteran (423k points) | 22 views

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3

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 in Linear Algebra by Arjun Veteran (423k points) | 13 views

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4

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 in Linear Algebra by Arjun Veteran (423k points) | 7 views

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1 answer

5

ISI2015-MMA-39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are $1,1,4$ $1,4,4$ $0,1,4$ $0,4,4$

asked Sep 23 in Linear Algebra by Arjun Veteran (423k points) | 11 views

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6

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 in Linear Algebra by Arjun Veteran (423k points) | 12 views

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7

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 in Linear Algebra by Arjun Veteran (423k points) | 11 views

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1 answer

8

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 $A=CB$

asked Sep 23 in Linear Algebra by Arjun Veteran (423k points) | 8 views

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1 answer

9

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 $4$ $5$ $6$ $7$

asked Sep 23 in Linear Algebra by Arjun Veteran (423k points) | 11 views

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1 answer

10

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 $A^{2010}$ ... $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

asked Sep 23 in Linear Algebra by Arjun Veteran (423k points) | 9 views

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1 answer

11

ISI2015-DCG-5
If $f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then the value of $\big(f(x)\big)^2$ is $f(x)$ $f(2x)$ $2f(x)$ None of these

asked Sep 18 in Linear Algebra by gatecse Boss (16.6k points) | 23 views

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1 answer

12

ISI2015-DCG-31
Let $A$ be an $n \times n$ matrix such that $\mid A^{2} \mid\: =1$. Here $\mid A \mid $ stands for determinant of matrix $A$. Then $\mid A \mid =1$ $\mid A \mid =0 \text{ or } 1$ $\mid A \mid =-1, 0 \text{ or } 1$ $\mid A \mid =-1 \text{ or } 1$

asked Sep 18 in Linear Algebra by gatecse Boss (16.6k points) | 17 views

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1 answer

13

ISI2015-DCG-32
The set of vectors constituting an orthogonal basis in $\mathbb{R} ^3$ is $\begin{Bmatrix} \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}, & \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, & \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \end{Bmatrix}$ ... None of these

asked Sep 18 in Linear Algebra by gatecse Boss (16.6k points) | 40 views

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1 answer

14

ISI2015-DCG-33
Suppose $A$ and $B$ are orthogonal $n \times n$ matrices. Which of the following is also an orthogonal matrix? Assume that $O$ is the null matrix of order $n \times n$ and $I$ is the identity matrix of order $n$. $AB-BA$ $\begin{pmatrix} A & O \\ O & B \end{pmatrix}$ $\begin{pmatrix} A & I \\ I & B \end{pmatrix}$ $A^2 – B^2$

asked Sep 18 in Linear Algebra by gatecse Boss (16.6k points) | 11 views

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1 answer

15

ISI2015-DCG-34
Let $A_{ij}$ denote the minors of an $n \times n$ matrix $A$. What is the relationship between $\mid A_{ij} \mid $ and $\mid A_{ji} \mid $? They are always equal $\mid A_{ij} \mid = – \mid A _{ji} \mid \text{ if } i \neq j$ They are equal if $A$ is a symmetric matrix If $\mid A_{ij} \mid =0$ then $\mid A_{ji} \mid =0$

asked Sep 18 in Linear Algebra by gatecse Boss (16.6k points) | 11 views

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1 answer

16

ISI2017-DCG-4
If $A$ is a $3 \times 3$ matrix satisfying $A^3 – A^2 +A-I= O$ (where $O$ is the zero matrix and $I$ is the identity matrix) then the value of $A^4$ is $A$ $O$ $I$ none of these

asked Sep 18 in Linear Algebra by gatecse Boss (16.6k points) | 9 views

+1 vote

1 answer

17

CMI2019-B-6
Let $A$ be an $n\times n $ matrix of integers such that each row and each column is arranged in ascending order. We want to check whether a number $k$ appears in $A.$ If $k$ is present, we should report its position - that is, the row $i$ and ... $A.$ Justify the complexity of your algorithm. For both algorithms, describe a worst-case input where $k$ is present in $A.$

asked Sep 13 in Algorithms by gatecse Boss (16.6k points) | 13 views

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2 answers

18

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 ________

asked Jun 2 in Linear Algebra by srestha Veteran (117k points) | 169 views

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1 answer

19

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?

asked May 27 in Linear Algebra by srestha Veteran (117k points) | 140 views

+1 vote

1 answer

20

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 $A^2=A$.

asked May 12 in Linear Algebra by akash.dinkar12 Boss (41.8k points) | 68 views

+1 vote

0 answers

21

CSIR UGC NET

asked Apr 28 in Linear Algebra by Hirak Active (3.5k points) | 51 views

+2 votes

1 answer

22

Nullity of matrix
Nullity of a matrix = Total number columns – Rank of that matrix But how to calculate value of x when nullity is already given(1 in this case)

asked Jan 24 in Linear Algebra by Nandkishor3939 Active (1.3k points) | 153 views

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1 answer

23

Simple Determinant
how to prove determinant 1 and 2 are same by some row-column manipulation ,please Help??

asked Jan 19 in Linear Algebra by BHASHKAR (25 points) | 63 views

+1 vote

0 answers

24

#set theory #groups
Consider the set H of all 3 × 3 matrices of the type: $\begin{bmatrix} a&f&e\\ 0&b&d\\ 0&0&c\\ \end{bmatrix}$ where a, b, c, d, e and f are real numbers and $abc ≠ 0$. Under the matrix multiplication operation, the set H is: (a) a group (b) a monoid but not a group (c) a semigroup but not a monoid (d) neither a group nor a semigroup

asked Jan 5 in Set Theory & Algebra by Kunal Kadian Active (2.8k points) | 70 views

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25

Determinant
True and false IF A is 5*5 matrix then det(4$A^{3}$)=$4^{5}det(A^{3}))$ det($(4A)^{3}$)=$det(4^{3}A^{3}))$ i think both are true ??

asked Jan 4 in Mathematical Logic by Gurdeep Saini Boss (10.2k points) | 54 views

+2 votes

1 answer

26

Cases when LU decomposition is possible.
I had already visited below link but was unable to grasp it. https://math.stackexchange.com/questions/218770/when-does-a-square-matrix-have-an-lu-decomposition. I made some form of self-analysis, let me know if I am in the correct ... pivot, and if such cannot be fixed without a row shuffle operation, then LU decomposition is not possible. Am I correct?

asked Nov 18, 2018 in Linear Algebra by Ayush Upadhyaya Boss (27.2k points) | 248 views

+1 vote

0 answers

27

Zeal Test Series 2019: Linear Algebra - Matrices

asked Nov 17, 2018 in Linear Algebra by Prince Sindhiya Loyal (5.6k points) | 78 views

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0 answers

28

Virtual Gate Test Series: Linear Algebra - Eigen Value

asked Oct 16, 2018 in Linear Algebra by Prince Sindhiya Loyal (5.6k points) | 57 views

0 votes

0 answers

29

Virtual Gate Test Series: Linear Algebra - Eigen Values of a Unity Matrix

asked Oct 15, 2018 in Linear Algebra by Prince Sindhiya Loyal (5.6k points) | 81 views

+4 votes

5 answers

30

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

asked Sep 20, 2018 in Linear Algebra by aditi19 Active (5k points) | 285 views

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