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Recent questions and answers in Linear Algebra

6 votes
3 answers
1
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 $A^{-1}$ sums ... statement $(1)$ and $(2)$ are correct but not necessarily statement $(3)$ all the three statements $(1), (2),$ and $(3)$ are correct
answered 4 days ago in Linear Algebra Gyanu 908 views
1 vote
2 answers
2
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 $\lambda_1+\lambda_2+\lambda_3 = \sqrt{2}+1$, then the set of possible values of $t, \: – \pi \leq t < \pi$, is Empty set $\{ \frac{\pi}{4} \}$ $\{ – \frac{\pi}{4}, \frac{\pi}{4} \}$ $\{ – \frac{\pi}{3}, \frac{\pi}{3} \}$
answered Oct 12 in Linear Algebra ayush.5 262 views
1 vote
2 answers
3
Let $A$ be a real $2 \times 2$ matrix. If $5+3i$ is an eigenvalue of $A$, then $det(A)$ equals 4 equals 8 equals 16 cannot be determined from the given information
answered Oct 12 in Linear Algebra ayush.5 136 views
0 votes
3 answers
4
The value of $\begin{vmatrix} 1 & \log _x y & \log_x z \\ \log _y x & 1 & \log_y z \\ \log _z x & \log _z y & 1 \end{vmatrix}$ is $0$ $1$ $-1$ None of these
answered Oct 12 in Linear Algebra ayush.5 93 views
9 votes
3 answers
6
If $A$ is a skew symmetric matrix then $A^t$ is Diagonal matrix $A$ $0$ $-A$
answered Sep 21 in Linear Algebra shivam001 4.6k views
1 vote
2 answers
7
An $n \times n$ matrix is said to be tridiagonal if its entries $a_{ij}$ are zero except when $|i&minus;j| \leq 1$ for $1 \leq i, \: j \leq n$. Note that only $3n &minus; 2$ entries of a tridiagonal matrix are non-zero. Thus, an array $L$ of size $3n &minus; 2$ ... a tridiagonal matrix. Given $i, j$, write pseudo-code to store $a_{ij}$ in $L$, and get the value of $a_{ij}$ stored earlier in $L$.
answered Sep 14 in Linear Algebra indranil21 327 views
3 votes
1 answer
8
Let A be an nxn real matrix such that A^2=I and y be an n-dimensional vector. Then the linear system of equations AX=Y has A) No solution B) Unique Solution C) More than one but finitely many independent solutions D) infinitely many independent solutions
answered Sep 4 in Linear Algebra Amit Tiwari 5 695 views
5 votes
2 answers
9
Let $A$ be the $2 \times 2$ matrix $\begin{pmatrix} \sin\frac{\pi}{18}&-\sin \frac{4\pi}{9} \\ \sin \frac{4\pi}{9}&\sin \frac {\pi}{18} \end{pmatrix}$. Then the smallest number $n \in \mathbb{N}$ such that $A^{n}=1$ is. $3$ $9$ $18$ $27$
answered Aug 25 in Linear Algebra ankitgupta.1729 221 views
12 votes
6 answers
10
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 steepest ... 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 Aug 23 in Linear Algebra raving_sajeev 1.7k views
0 votes
1 answer
11
Consider three vectors $x=\begin{bmatrix}1\\2 \end{bmatrix}, y=\begin{bmatrix}4\\8 \end{bmatrix},z=\begin{bmatrix}3\\1 \end{bmatrix}$. Which of the folowing statements is true $x$ and $y$ are linearly independent $x$ and $y$ are linearly dependent $x$ and $z$ are linearly dependent $y$ and $z$ are linearly dependent
answered Aug 21 in Linear Algebra nocturnal123 42 views
0 votes
1 answer
12
Is the determinent of both the orthogonal and orthonormal is +-1?for orthonormal im always getting +-1 but for orthogonal its always +-C,not 1,C=mag of the vector matrix
answered Aug 18 in Linear Algebra Rish9799 191 views
19 votes
3 answers
13
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 Aug 15 in Linear Algebra Gyanu 1.7k views
0 votes
2 answers
14
Let A be a 3*3 matrix whose characteristics roots are 3,2,-1. If $B=A^2-A$ then |B|=? a)24 b)-2 c)12 d)-12 Please explain in detail.
answered Aug 11 in Linear Algebra AkashChandraGupta 814 views
22 votes
6 answers
15
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 Aug 11 in Linear Algebra Jhaiyam 5.7k views
1 vote
2 answers
16
$AVA=A$ is called : Identity law De Morgan’s law Idempotent law Complement law
answered Aug 8 in Linear Algebra AkashChandraGupta 78 views
0 votes
2 answers
17
20 votes
4 answers
18
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 Identity 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 Aug 8 in Linear Algebra AkashChandraGupta 1.8k views
7 votes
6 answers
19
Suppose the rank of the matrix $\begin{pmatrix}1&1&2&2\\1&1&1&3\\a&b&b&1\end{pmatrix}$ is $2$ for some real numbers $a$ and $b$. Then $b$ equals $1$ $3$ $1/2$ $1/3$
answered Aug 8 in Linear Algebra AkashChandraGupta 999 views
6 votes
2 answers
20
For vectors $x, \: y$ in $\mathbb{R}^n$, define the inner product $\langle x, y \rangle = \Sigma^n_{i=1} x_iy_i$, and the length of $x$ to be $\| x \| = \sqrt{\langle x, x \rangle}$. Let $a, \: b$ be two vectors in $\mathbb{R} ^n$ ... $a, \: b$? Choose from the following options. ii only i and ii iii only iv only iv and v
answered Jul 4 in Linear Algebra arks 453 views
52 votes
5 answers
21
Let $A$ be $n\times n$ real valued square symmetric matrix of rank 2 with $\sum_{i=1}^{n}\sum_{j=1}^{n}A^{2}_{ij} =$ 50. Consider the following statements. One eigenvalue must be in $\left [ -5,5 \right ]$ The eigenvalue with the largest ... strictly greater than 5 Which of the above statements about eigenvalues of $A$ is/are necessarily CORRECT? Both I and II I only II only Neither I nor II
answered Jul 2 in Linear Algebra arks 9.7k views
1 vote
1 answer
22
Consider following system of equations: $\begin{bmatrix} 1 &2 &3 &4 \\ 5&6 &7 &8 \\ a&9 &b &10 \\ 6&8 &10 & 13 \end{bmatrix}$\begin{bmatrix} x1\\ x2\\ x3\\ x4 \end{bmatrix}$=$\begin{bmatrix} 0\\ 0\\ 0\\ 0 \end{bmatrix}$ The locus of all $(a,b)\in\mathbb{R}^{2 ... system has at least two distinct solution for ($x_{1},x_{2},x_{3},x_{4}$) is a parabola a straight line entire $\mathbb{R}^{2}$ a point
answered Jun 8 in Linear Algebra Amartya 458 views
0 votes
2 answers
23
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 $P_3$ intersect ... 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 May 18 in Linear Algebra Amartya 190 views
0 votes
1 answer
24
$M$ is a square matrix of order $’n’$ and its determinant value is $5.$ If all the elements of $M$ are multiplied by $2,$ its determinant value becomes $40.$ The value of $’n’$ is $2$ $3$ $5$ $4$
answered May 3 in Linear Algebra Mohit Kumar 6 88 views
11 votes
6 answers
25
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 30 in Linear Algebra Ajay_singh 3.6k views
0 votes
2 answers
26
19 votes
5 answers
27
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 Apr 21 in Linear Algebra DIBAKAR MAJEE 3.2k views
17 votes
4 answers
28
If matrix $X = \begin{bmatrix} a & 1 \\ -a^2+a-1 & 1-a \end{bmatrix}$ and $X^2 - X + I = O$ ($I$ is the identity matrix and $O$ is the zero matrix), then the inverse of $X$ is $\begin{bmatrix} 1-a &-1 \\ a^2& a \end{bmatrix}$ $\begin{bmatrix} 1-a &-1 \\ a^2-a+1& a \end{bmatrix}$ $\begin{bmatrix} -a &1 \\ -a^2+a-1& 1-a \end{bmatrix}$ $\begin{bmatrix} a^2-a+1 &a \\ 1& 1-a \end{bmatrix}$
answered Apr 21 in Linear Algebra DIBAKAR MAJEE 1.6k views
14 votes
4 answers
29
What values of x, y and z satisfy the following system of linear equations? $\begin{bmatrix} 1 &2 &3 \\ 1& 3 &4 \\ 2& 2 &3 \end{bmatrix} \begin{bmatrix} x\\y \\ z \end{bmatrix} = \begin{bmatrix} 6\\8 \\ 12 \end{bmatrix}$ $x = 6$, $y = 3$, $z = 2$ $x = 12$, $y = 3$, $z = - 4$ $x = 6$, $y = 6$, $z = - 4$ $x = 12$, $y = - 3$, $z = 0$
answered Apr 21 in Linear Algebra DIBAKAR MAJEE 2k views
18 votes
7 answers
30
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 Apr 21 in Linear Algebra DIBAKAR MAJEE 2.5k views
64 votes
8 answers
31
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 Apr 20 in Linear Algebra Ashish Patel 15 15.3k views
0 votes
1 answer
32
0 votes
0 answers
33
If product of matrix $A=\begin{bmatrix}\cos^{2}\theta &\cos \theta \sin \theta \\ \cos \theta \sin \theta &\sin ^{2} \theta& \end{bmatrix}$ and $B=\begin{bmatrix}\cos^{2}\phi &\cos \phi \sin \phi \\ \cos \phi \sin \phi &\sin ^{2} \phi& \end{bmatrix}$ is a ... and $\phi$ differ by an odd multiple of $\pi$ even multiple of $\pi$ odd multiple of $\dfrac{\pi}{2}$ even multiple of $\dfrac{\pi}{2}$
asked Apr 2 in Linear Algebra Lakshman Patel RJIT 37 views
0 votes
1 answer
36
0 votes
1 answer
37
Consider two matrices $M_1$ and $M_2$ with $M_1^*M_2=0$ and $M_1$ is non singular. Then which of the following is true? $M_2$ is non singular $M_2$ is null matrix $M_2$ is the identity matrix $M_2$ is transpose of $M_1$
asked Mar 30 in Linear Algebra Lakshman Patel RJIT 97 views
2 votes
1 answer
38
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 ankitgupta.1729 135 views
1 vote
1 answer
39
5 votes
3 answers
40
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 smsubham 328 views
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