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Recent questions and answers in Linear Algebra
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Recent questions and answers in Linear Algebra
7
votes
5
answers
1
ISI2017-MMA-29
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 $1$ $3$ $1/2$ $1/3$
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
Jul 5
in
Linear Algebra
Shankhajit Roy
727
views
isi2017-mma
engineering-mathematics
linear-algebra
rank-of-matrix
4
votes
2
answers
2
TIFR2017-A-2
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
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
364
views
tifr2017
linear-algebra
vector-space
20
votes
3
answers
3
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 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
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
Jul 2
in
Linear Algebra
arks
1.5k
views
gate1997
linear-algebra
easy
matrices
50
votes
5
answers
4
GATE2017-1-31
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 ... 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
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
8.6k
views
gate2017-1
linear-algebra
eigen-value
normal
0
votes
1
answer
5
NIELIT 2016 MAR Scientist B - Section B: 12
If $A$ and $B$ are square matrices of size $n\times n$, then which of the following statements is not true? $\det(AB)=\det(A) \det(B)$ $\det(kA)=k^n \det(A)$ $\det(A+B)=\det(A)+\det(B)$ $\det(A^T)=1/\det(A^{-1})$
If $A$ and $B$ are square matrices of size $n\times n$, then which of the following statements is not true? $\det(AB)=\det(A) \det(B)$ $\det(kA)=k^n \det(A)$ $\det(A+B)=\det(A)+\det(B)$ $\det(A^T)=1/\det(A^{-1})$
answered
Jun 12
in
Linear Algebra
palashbehra5
81
views
nielit2016mar-scientistb
1
vote
1
answer
6
UGCNET-Dec2004-II: 1
$AVA=A$ is called : Identity law De Morgan’s law Idempotent law Complement law
$AVA=A$ is called : Identity law De Morgan’s law Idempotent law Complement law
answered
Jun 12
in
Linear Algebra
palashbehra5
56
views
ugcnetdec2004ii
0
votes
1
answer
7
ISI2017-MMA-18
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 \ ... 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
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
313
views
isi2017-mma
engineering-mathematics
linear-algebra
system-of-equations
0
votes
1
answer
8
NIELIT 2016 DEC Scientist B (IT) - Section B: 26
Two eigenvalues of a $3\times3$ real matrix $P$ are $(2+ \sqrt-1)$ and $3$. The determinant of $P$ is ________. $0$ $1$ $15$ $-1$
Two eigenvalues of a $3\times3$ real matrix $P$ are $(2+ \sqrt-1)$ and $3$. The determinant of $P$ is ________. $0$ $1$ $15$ $-1$
answered
May 27
in
Linear Algebra
Mohit Kumar 6
55
views
nielit2016dec-scientistb-it
0
votes
2
answers
9
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
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
124
views
isi2015-mma
linear-algebra
system-of-equations
10
votes
6
answers
10
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
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
3.1k
views
gate2019
engineering-mathematics
linear-algebra
determinant
19
votes
5
answers
11
GATE2005-IT-3
The determinant of the matrix given below is $\begin{bmatrix} 0 &1 &0 &2 \\ -1& 1& 1& 3\\ 0&0 &0 & 1\\ 1& -2& 0& 1 \end{bmatrix}$ $-1$ $0$ $1$ $2$
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
2.8k
views
gate2005-it
linear-algebra
normal
determinant
17
votes
4
answers
12
GATE2004-IT-36
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 $\begin{bmatrix} 1-a &-1 \\ a^2& a \end{bmatrix}$ ... $\begin{bmatrix} -a &1 \\ -a^2+a-1& 1-a \end{bmatrix}$ $\begin{bmatrix} a^2-a+1 &a \\ 1& 1-a \end{bmatrix}$
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
1.4k
views
gate2004-it
linear-algebra
matrices
normal
13
votes
4
answers
13
GATE2004-IT-6
What values of x, y and z satisfy the following system of linear equations? $\begin{bmatrix} 1 &2 &3 \\ 1& 3 &4 \\ 2& 2 &3 \end{bmatrix} \begin{bmatrix} x\\y \\ z \end{bmatrix} = \begin{bmatrix} 6\\8 \\ 12 \end{bmatrix}$ $x = 6$, $y = 3$, $z = 2$ $x = 12$, $y = 3$, $z = - 4$ $x = 6$, $y = 6$, $z = - 4$ $x = 12$, $y = - 3$, $z = 0$
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
1.6k
views
gate2004-it
linear-algebra
system-of-equations
easy
17
votes
7
answers
14
GATE2006-IT-26
What are the eigenvalues of the matrix $P$ given below $P= \begin{pmatrix} a &1 &0 \\ 1& a& 1\\ 0&1 &a \end{pmatrix}$ $a, a -√2, a + √2$ $a, a, a$ $0, a, 2a$ $-a, 2a, 2a$
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
2.2k
views
gate2006-it
linear-algebra
eigen-value
normal
61
votes
8
answers
15
GATE2014-2-47
The product of the non-zero eigenvalues of the matrix is ____ $\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}$
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
13.7k
views
gate2014-2
linear-algebra
eigen-value
normal
numerical-answers
0
votes
1
answer
16
NIELIT 2016 DEC Scientist B (CS) - Section B: 21
Let $A,B,C,D$ be $n\times n$ matrices, each with non-zero determinant. If $ABCD=1$, then $B^{-1}$ is: $D^{-1}C^{-1}A^{-1}$ $CDA$ $ADC$ Does not necessarily exist.
Let $A,B,C,D$ be $n\times n$ matrices, each with non-zero determinant. If $ABCD=1$, then $B^{-1}$ is: $D^{-1}C^{-1}A^{-1}$ $CDA$ $ADC$ Does not necessarily exist.
answered
Apr 1
in
Linear Algebra
haralk10
60
views
nielit2016dec-scientistb-cs
0
votes
1
answer
17
NIELIT 2016 MAR Scientist B - Section B: 4
What is the determinant of the matrix $\begin{bmatrix}5&3&2\\1&2&6\\3&5&10\end{bmatrix}$ $-76$ $-28$ $+28$ $+72$
What is the determinant of the matrix $\begin{bmatrix}5&3&2\\1&2&6\\3&5&10\end{bmatrix}$ $-76$ $-28$ $+28$ $+72$
answered
Mar 31
in
Linear Algebra
Ashwani Kumar 2
94
views
nielit2016mar-scientistb
0
votes
1
answer
18
NIELIT 2016 MAR Scientist B - Section B: 6
The system of simultaneous equations $x+2y+z=6\\2x+y+2z=6\\x+y+z=5$ has unique solution. infinite number of solutions. no solution. exactly two solutions.
The system of simultaneous equations $x+2y+z=6\\2x+y+2z=6\\x+y+z=5$ has unique solution. infinite number of solutions. no solution. exactly two solutions.
answered
Mar 31
in
Linear Algebra
haralk10
74
views
nielit2016mar-scientistb
0
votes
1
answer
19
NIELIT 2017 DEC Scientist B - Section B: 60
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 $M_1$
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
44
views
nielit2017dec-scientistb
1
vote
1
answer
20
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 $\lambda_1+\lambda_2+\lambda_3 = \sqrt{2}+1$ ... $\{ - \frac{\pi}{4}, \frac{\pi}{4} \}$ $\{ - \frac{\pi}{3}, \frac{\pi}{3} \}$
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
Mar 14
in
Linear Algebra
haralk10
156
views
isi2015-mma
linear-algebra
matrices
eigen-value
1
vote
1
answer
21
TIFR2020-A-2
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
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
92
views
tifr2020
engineering-mathematics
linear-algebra
rank-of-matrix
eigen-value
1
vote
1
answer
22
GF MT 4
Is the answer and explaination given correct ?
Is the answer and explaination given correct ?
answered
Mar 4
in
Linear Algebra
smsubham
183
views
gateforum-test-series
linear-algebra
rank-of-matrix
5
votes
3
answers
23
ISI2015-MMA-39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are $1,1,4$ $1,4,4$ $0,1,4$ $0,4,4$
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
232
views
isi2015-mma
linear-algebra
matrices
eigen-value
4
votes
3
answers
24
GATE2020-CS-27
Let $A$ and $B$ be two $n \times n$ matrices over real numbers. Let rank($M$) and $\text{det}(M)$ denote the rank and determinant of a matrix $M$, respectively. Consider the following statements. $\text{rank}(AB) = \text{rank }(A) \text{rank }(B)$ ... Which of the above statements are TRUE? I and II only I and IV only II and III only III and IV only
Let $A$ and $B$ be two $n \times n$ matrices over real numbers. Let rank($M$) and $\text{det}(M)$ denote the rank and determinant of a matrix $M$, respectively. Consider the following statements. $\text{rank}(AB) = \text{rank }(A) \text{rank }(B)$ ... Which of the above statements are TRUE? I and II only I and IV only II and III only III and IV only
answered
Feb 26
in
Linear Algebra
immanujs
1.2k
views
gate2020-cs
discrete-mathematics
engineering-mathematics
matrices
5
votes
5
answers
25
GATE2020-CS-18
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
answered
Feb 26
in
Linear Algebra
immanujs
1.5k
views
gate2020-cs
numerical-answers
engineering-mathematics
group-theory
0
votes
0
answers
26
Introduction to Linear Algebra 4th edition Problem Set 1.1
How many corner does a cube have in 4 dimensions? How many 3D faces? Now by observation we can tell that, an n-dimensional cube has $2^n$ corners. 1D cube which is a line have $2^1$ corners 2D cube which is a square have $2^2$ ... . but this is the question i'm not able to answer. How every N-cube have $|2n|$ cubes of dimension (N-1)?
How many corner does a cube have in 4 dimensions? How many 3D faces? Now by observation we can tell that, an n-dimensional cube has $2^n$ corners. 1D cube which is a line have $2^1$ corners 2D cube which is a square have $2^2$ corners 3D cube have $2^3$ corners ... 8 three-dimension cubes. but this is the question i'm not able to answer. How every N-cube have $|2n|$ cubes of dimension (N-1)?
asked
Feb 26
in
Linear Algebra
Mk Utkarsh
137
views
linear-algebra
0
votes
1
answer
27
TIFR2020-A-12
The hour needle of a clock is malfunctioning and travels in the anti-clockwise direction, i.e., opposite to the usual direction, at the same speed it would have if it was working correctly. The minute needle is working correctly. Suppose the two needles show the correct time at $12$ ... $\dfrac{11}{12}$ hour $\dfrac{12}{13}$ hour $\dfrac{19}{22}$ hour One hour
The hour needle of a clock is malfunctioning and travels in the anti-clockwise direction, i.e., opposite to the usual direction, at the same speed it would have if it was working correctly. The minute needle is working correctly. Suppose the two needles show the correct time at $12$ noon, thus ... ? $\dfrac{10}{11}$ hour $\dfrac{11}{12}$ hour $\dfrac{12}{13}$ hour $\dfrac{19}{22}$ hour One hour
answered
Feb 19
in
Linear Algebra
ankitgupta.1729
71
views
tifr2020
0
votes
1
answer
28
TIFR2020-A-5
Let $A$ be am $n\times n$ invertible matrix with real entries whose column sums are all equal to $1$. Consider the following statements: Every column in the matrix $A^{2}$ sums to $2$ Every column in the matrix $A^{3}$ sums to $3$ Every column in the matrix $A^{-1}$ ... statement $(3)$ is correct but not statements $(1)$ or $(2)$ all the $3$ statements $(1),(2),$ and $(3)$ are correct
Let $A$ be am $n\times n$ invertible matrix with real entries whose column sums are all equal to $1$. Consider the following statements: Every column in the matrix $A^{2}$ sums to $2$ Every column in the matrix $A^{3}$ sums to $3$ Every column in the matrix $A^{-1}$ ... $(3)$ is correct but not statements $(1)$ or $(2)$ all the $3$ statements $(1),(2),$ and $(3)$ are correct
answered
Feb 11
in
Linear Algebra
Lakshman Patel RJIT
127
views
tifr2020
engineering-mathematics
linear-algebra
matrices
0
votes
0
answers
29
TIFR2020-A-3
Let $d\geq 4$ and fix $w\in \mathbb{R}.$ Let $S = \{a = (a_{0},a_{1},\dots ,a_{d})\in \mathbb{R}^{d+1}\mid f_{a}(w) = 0\: \text{and}\: f'_{a}(w) = 0\},$ where the polynomial function $f_{a}(x)$ ... is a $d$-dimensional vector subspace of $\mathbb{R}^{d+1}$ $S$ is a $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1}$ None of the other options
Let $d\geq 4$ and fix $w\in \mathbb{R}.$ Let $S = \{a = (a_{0},a_{1},\dots ,a_{d})\in \mathbb{R}^{d+1}\mid f_{a}(w) = 0\: \text{and}\: f'_{a}(w) = 0\},$ where the polynomial function $f_{a}(x)$ ... $d$-dimensional vector subspace of $\mathbb{R}^{d+1}$ $S$ is a $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1}$ None of the other options
asked
Feb 10
in
Linear Algebra
Lakshman Patel RJIT
65
views
tifr2020
engineering-mathematics
linear-algebra
vector-space
0
votes
2
answers
30
A is a 4-square matrix and A 5 = 0. Then
A is a 4-square matrix and A_5 (a raised to the power of 5) = 0. Then A_4 = a) I b) -I c) 0 d) A
A is a 4-square matrix and A_5 (a raised to the power of 5) = 0. Then A_4 = a) I b) -I c) 0 d) A
answered
Jan 31
in
Linear Algebra
smsubham
1.2k
views
matrices
linear-algebra
38
votes
5
answers
31
GATE2018-26
Consider a matrix P whose only eigenvectors are the multiples of $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$. Consider the following statements. P does not have an inverse P has a repeated eigenvalue P cannot be diagonalized Which one of the following options ... and III are necessarily true Only II is necessarily true Only I and II are necessarily true Only II and III are necessarily true
Consider a matrix P whose only eigenvectors are the multiples of $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$. Consider the following statements. P does not have an inverse P has a repeated eigenvalue P cannot be diagonalized Which one of the following options is correct ... I and III are necessarily true Only II is necessarily true Only I and II are necessarily true Only II and III are necessarily true
answered
Jan 29
in
Linear Algebra
JashanArora
8.6k
views
gate2018
linear-algebra
matrices
eigen-value
normal
37
votes
4
answers
32
GATE2016-2-04
Consider the system, each consisting of $m$ linear equations in $n$ variables. If $m < n$, then all such systems have a solution. If $m > n$, then none of these systems has a solution. If $m = n$, then there exists a system which has a solution. Which one of the ... is CORRECT? $I, II$ and $III$ are true. Only $II$ and $III$ are true. Only $III$ is true. None of them is true.
Consider the system, each consisting of $m$ linear equations in $n$ variables. If $m < n$, then all such systems have a solution. If $m > n$, then none of these systems has a solution. If $m = n$, then there exists a system which has a solution. Which one of the following is CORRECT? $I, II$ and $III$ are true. Only $II$ and $III$ are true. Only $III$ is true. None of them is true.
answered
Jan 27
in
Linear Algebra
JashanArora
5.5k
views
gate2016-2
linear-algebra
system-of-equations
normal
19
votes
2
answers
33
TIFR2012-B-12
Let $A$ be a matrix such that $A^{k}=0$. What is the inverse of $I - A$? $0$ $I$ $A$ $1 + A + A^{2} + ...+ A^{k - 1}$ Inverse is not guaranteed to exist.
Let $A$ be a matrix such that $A^{k}=0$. What is the inverse of $I - A$? $0$ $I$ $A$ $1 + A + A^{2} + ...+ A^{k - 1}$ Inverse is not guaranteed to exist.
answered
Jan 26
in
Linear Algebra
blackcloud
1.1k
views
tifr2012
linear-algebra
matrices
39
votes
7
answers
34
GATE2007-IT-2
Let $A$ be the matrix $\begin{bmatrix}3 &1 \\ 1&2\end{bmatrix}$. What is the maximum value of $x^TAx$ where the maximum is taken over all $x$ that are the unit eigenvectors of $A?$ $5$ $\frac{(5 + √5)}{2}$ $3$ $\frac{(5 - √5)}{2}$
Let $A$ be the matrix $\begin{bmatrix}3 &1 \\ 1&2\end{bmatrix}$. What is the maximum value of $x^TAx$ where the maximum is taken over all $x$ that are the unit eigenvectors of $A?$ $5$ $\frac{(5 + √5)}{2}$ $3$ $\frac{(5 - √5)}{2}$
answered
Jan 26
in
Linear Algebra
blackcloud
4.7k
views
gate2007-it
linear-algebra
eigen-value
normal
2
votes
2
answers
35
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?
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 matrix is ... under 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
Sahin
448
views
group-theory
16
votes
3
answers
36
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
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 $\begin{bmatrix} 2& 1& 1& 1\\ 1& 2& 1& 1\\ 1& 1& 2& 1\\ 1& 1& 1& 2 \end{bmatrix}$ A) 2 B) 5 C) 8 D) 16
answered
Jan 4
in
Linear Algebra
Satbir
2k
views
gate2013-ec
linear-algebra
engineering-mathematics
normal
determinant
11
votes
4
answers
37
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 _______
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
ankitgupta.1729
5.1k
views
gate2019
numerical-answers
engineering-mathematics
linear-algebra
eigen-value
21
votes
5
answers
38
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 $2^X$) $\text{power} \left(2, n\right)$ $\text{power} \left(2, n^2\right)$ $\text{power} \left(2,\frac{ \left(n^2+ n \right) }{2}\right)$ $\text{power} \left(2, \frac{\left(n^2 - n\right)}{2}\right)$
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
JashanArora
4.9k
views
gate2004
linear-algebra
normal
matrices
0
votes
1
answer
39
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 $\pi$ $\frac{\pi}{2}$ $0$ $1$
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
ajaysoni1924
80
views
isi2017-dcg
linear-algebra
determinant
definite-integrals
non-gate
1
vote
1
answer
40
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 $1$ $2$ $3$ $4$
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
ajaysoni1924
141
views
isi2015-mma
linear-algebra
determinant
functions
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