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Recent questions and answers in Linear Algebra
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Linear Dependency
I have a confusion, let's say we have three vectors a,b & c in 2D space. a & b are linearly independent but c is dependent on a & b. Can't we represent whole 2D space using a,b & c as (any vector V) V = x*a + y*b +0*c and hence a,b & c form basis. Then how can we say that basis are set of linearly independent vectors. Please tell me where I am going wrong.
asked
5 days
ago
in
Linear Algebra
by
Codered03
(
5
points)

5
views
enggmaths
0
votes
0
answers
TIFRGS2021 Question
Let $M$ be a $n\times m$ real matrix. Consider the following: Let $k_1$ be the smallest number such that $M$ can be factorized as $A.B$, where $A$ is an $n\times k_1$ matrix and $B$ is a $k_1\times m$ matrix. Let $k_2$ ... $k_2 = k_3 < k_1$ (D) $k_1 = k_2 = k_3$ (E) No general relationship exists among $k_1$, $k_2$ and $k_3$
asked
Mar 24
in
Linear Algebra
by
zxy123
(
3.6k
points)

6
views
tifr2021
matrices
0
votes
0
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Gate overflow book
https://gateoverflow.in/204100/gate201826. The above is the link for the question. I am not able to understand the explaination for statement I i.e. the matrix may have an inverse. Can someone provide a proof or a counter example for the specific situation when the eigenvectors are multiples of a single vector the matrix may or may not have an inverse?
asked
Jan 19
in
Linear Algebra
by
hadarsh
(
5
points)

14
views
linearalgebra
+1
vote
0
answers
Testbook test series linear algebra
Let M be a real 4X4 matrix. Consider the following: s1: M has 4 linearly independent eigenvectors s2: M has 4 distinct eigenvalues s3: M is non singular (invertible) Which of the following is true? s1 implies s2 s1 implies s3 s2 implies s1 s3 implies s2 The answer given was s2 implies s1 which is fine. But why not also s1 implies s2?
asked
Dec 10, 2020
in
Linear Algebra
by
Dheera
(
14
points)

41
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linearalgebra
0
votes
0
answers
No. of nonzero eigenvalues <= Rank of the matrix
Does anyone knows the proof of this concept, I am not able to find it. Reference question – https://gateoverflow.in/2013/gate2014247
asked
Nov 27, 2020
in
Linear Algebra
by
smn98
(
5
points)

38
views
linearalgebra
enggmaths
linearalgebra
0
votes
1
answer
Idempotent Matrix
IF A and B are two matrices such that A.B = A, Will B always be an Identity matrix?
answered
Aug 16, 2020
in
Linear Algebra
by
Arkaprava
(
801
points)

36
views
0
votes
1
answer
Which of the following must be an eigenvector ofA?
answered
Aug 15, 2020
in
Linear Algebra
by
Arkaprava
(
801
points)

22
views
linearalgebra
engi
+1
vote
1
answer
GATE2020CS18 Video Solution
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
answered
Apr 25, 2020
in
Linear Algebra
by
amitkhurana512
(
47
points)

33
views
gate2020cs
numericalanswers
engineeringmathematics
grouptheory
videosolution
0
votes
0
answers
GATE2014247 Video Solution
The product of the nonzero eigenvalues of the matrix is ____ $\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}$
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

20
views
gate20142
linearalgebra
eigenvalue
normal
numericalanswers
videosolution
0
votes
0
answers
GATE201826 Video Solution
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 ... III are necessarily true Only II is necessarily true Only I and II are necessarily true Only II and III are necessarily true
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

14
views
gate2018
linearalgebra
matrices
eigenvalue
normal
videosolution
0
votes
0
answers
GATE2017131 Video Solution
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 ... 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
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

8
views
gate20171
linearalgebra
eigenvalue
normal
videosolution
0
votes
0
answers
GATE201713 Video Solution
Let $c_{1}.....c_{n}$ be scalars, not all zero, such that $\sum_{i=1}^{n}c_{i}a_{i}$ = 0 where $a_{i}$ are column vectors in $R^{n}$. Consider the set of linear equations $Ax = b$ ... has a unique solution at $x=J_{n}$ where $J_{n}$ denotes a $n$dimensional vector of all 1. no solution infinitely many solutions finitely many solutions
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

7
views
gate20171
linearalgebra
systemofequations
normal
videosolution
0
votes
0
answers
GATE2017130 Video Solution
Let $u$ and $v$ be two vectors in R2 whose Euclidean norms satisfy $\left \ u \right \ = 2\left \ v \right \$. What is the value of $\alpha$ such that $w = u + \alpha v$ bisects the angle between $u$ and $v$? $2$ $\frac{1}{2}$ $1$ $\frac{ 1}{2}$
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

13
views
gate20171
linearalgebra
normal
vectorspace
videosolution
0
votes
0
answers
GATE201415 Video Solution
The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a $4by4$ symmetric positive definite matrix is ___________
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

10
views
gate20141
linearalgebra
eigenvalue
numericalanswers
normal
videosolution
0
votes
0
answers
GATE2016204 Video Solution
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 ... CORRECT? $I, II$ and $III$ are true. Only $II$ and $III$ are true. Only $III$ is true. None of them is true.
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

13
views
gate20162
linearalgebra
systemofequations
normal
videosolution
0
votes
0
answers
GATE201944 Video Solution
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 _______
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

11
views
gate2019
numericalanswers
engineeringmathematics
linearalgebra
eigenvalue
videosolution
0
votes
0
answers
GATE2017252 Video Solution
If the characteristic polynomial of a 3 $\times$ 3 matrix $M$ over $\mathbb{R}$ (the set of real numbers) is $\lambda^3 – 4 \lambda^2 + a \lambda +30, \quad a \in \mathbb{R}$, and one eigenvalue of $M$ is 2, then the largest among the absolute values of the eigenvalues of $M$ is _______
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

24
views
gate20172
engineeringmathematics
linearalgebra
numericalanswers
eigenvalue
videosolution
0
votes
0
answers
GATE2016105 Video Solution
Two eigenvalues of a $3 \times 3$ real matrix $P$ are $(2+\sqrt {1})$ and $3$. The determinant of $P$ is _______
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

11
views
gate20161
linearalgebra
eigenvalue
numericalanswers
normal
videosolution
0
votes
0
answers
GATE2007IT2 Video Solution
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 $A?$ $5$ $\frac{(5 + √5)}{2}$ $3$ $\frac{(5  √5)}{2}$
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

4
views
gate2007it
linearalgebra
eigenvalue
normal
videosolution
0
votes
0
answers
GATE200426 Video Solution
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 $2^X$) $\text{power} \left(2, n\right)$ $\text{power} \left(2, n^2\right)$ $\text{power} \left(2,\frac{ \left(n^2+ n \right) }{2}\right)$ $\text{power} \left(2, \frac{\left(n^2  n\right)}{2}\right)$
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

10
views
gate2004
linearalgebra
normal
matrices
videosolution
0
votes
0
answers
GATE200727 Video Solution
Consider the set of (column) vectors defined by$X = \left \{x \in R^3 \mid x_1 + x_2 + x_3 = 0, \text{ where } x^T = \left[x_1,x_2,x_3\right]^T\right \}$ ... 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
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

7
views
gate2007
linearalgebra
normal
vectorspace
videosolution
0
votes
0
answers
GATE201414 Video Solution
Consider the following system of equations: $3x + 2y = 1 $ $4x + 7z = 1 $ $x + y + z = 3$ $x  2y + 7z = 0$ The number of solutions for this system is ______________
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

7
views
gate20141
linearalgebra
systemofequations
numericalanswers
normal
videosolution
0
votes
0
answers
GATE200725 Video Solution
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? $5$ $7$ $2$ $1$
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

8
views
gate2007
eigenvalue
linearalgebra
difficult
videosolution
0
votes
0
answers
GATE200341 Video Solution
Consider the following system of linear equations ... linearly dependent. For how many values of $\alpha$, does this system of equations have infinitely many solutions? \(0\) \(1\) \(2\) \(3\)
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

7
views
gate2003
linearalgebra
systemofequations
normal
videosolution
0
votes
0
answers
GATE2015118 Video Solution
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_________________.
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

9
views
gate20151
linearalgebra
matrices
numericalanswers
videosolution
0
votes
0
answers
GATE2015333 Video Solution
If the following system has nontrivial solution, $px + qy + rz = 0$ $qx + ry + pz = 0$ $rx + py + qz = 0$, then which one of the following options is TRUE? $p  q + r = 0 \text{ or } p = q = r$ $p + q  r = 0 \text{ or } p = q = r$ $p + q + r = 0 \text{ or } p = q = r$ $p  q + r = 0 \text{ or } p = q = r$
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

7
views
gate20153
linearalgebra
systemofequations
normal
videosolution
0
votes
0
answers
GATE2017222 Video Solution
Let $P = \begin{bmatrix}1 & 1 & 1 \\2 & 3 & 4 \\3 & 2 & 3\end{bmatrix}$ and $Q = \begin{bmatrix}1 & 2 &1 \\6 & 12 & 6 \\5 & 10 & 5\end{bmatrix}$ be two matrices. Then the rank of $ P+Q$ is ___________ .
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

6
views
gate20172
linearalgebra
eigenvalue
numericalanswers
videosolution
0
votes
0
answers
GATE2016206 Video Solution
Suppose that the eigenvalues of matrix $A$ are $1, 2, 4$. The determinant of $\left(A^{1}\right)^{T}$ is _________.
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

7
views
gate20162
linearalgebra
eigenvalue
normal
numericalanswers
videosolution
0
votes
0
answers
GATE19961.7 Video Solution
Let $Ax = b$ be a system of linear equations where $A$ is an $m \times n$ matrix and $b$ is a $m \times 1$ column vector and $X$ is an $n \times1$ column vector of unknowns. Which of the following is false? The system has a solution if and only ... a unique solution. The system will have only a trivial solution when $m=n$, $b$ is the zero vector and $\text{rank}(A) =n$.
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

7
views
gate1996
linearalgebra
systemofequations
normal
videosolution
0
votes
0
answers
GATE200427 Video Solution
Let $A, B, C, D$ be $n \times n$ matrices, each with nonzero determinant. If $ABCD = I$, then $B^{1}$ is $D^{1}C^{1}A^{1}$ $CDA$ $ADC$ Does not necessarily exist
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

7
views
gate2004
linearalgebra
normal
matrices
videosolution
0
votes
0
answers
GATE2015315 Video Solution
In the given matrix $\begin{bmatrix} 1 & 1 & 2 \\ 0 & 1 & 0 \\ 1 & 2 & 1 \end{bmatrix}$ ... $\left\{a\left( \sqrt{2},0,1\right) \mid a \neq 0, a \in \mathbb{R}\right\}$
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

7
views
gate20153
linearalgebra
eigenvalue
normal
videosolution
0
votes
0
answers
GATE200476 Video Solution
In an $M \times N$ matrix all nonzero entries are covered in $a$ rows and $b$ columns. Then the maximum number of nonzero entries, such that no two are on the same row or column, is $\leq a +b$ $\leq \max(a, b)$ $\leq \min(Ma, Nb)$ $\leq \min(a, b)$
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

9
views
gate2004
linearalgebra
normal
matrices
videosolution
0
votes
0
answers
GATE201817 Video Solution
Consider a matrix $A= uv^T$ where $u=\begin{pmatrix}1 \\ 2 \end{pmatrix} , v = \begin{pmatrix}1 \\1 \end{pmatrix}$. Note that $v^T$ denotes the transpose of $v$. The largest eigenvalue of $A$ is ____
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

6
views
gate2018
linearalgebra
eigenvalue
normal
numericalanswers
videosolution
0
votes
0
answers
GATE201211 Video Solution
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 $−512 \sqrt{2}$
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

6
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gate2012
linearalgebra
eigenvalue
videosolution
0
votes
0
answers
GATE20083 Video Solution
The following system of equations $x_1 + x_2 + 2x_3 = 1$ $x_1 + 2x_2 + 3x_3 = 2$ $x_1 + 4x_2 + αx_3 = 4$ has a unique solution. The only possible value(s) for $α$ is/are $0$ either $0$ or $1$ one of $0, 1$, or $1$ any real number
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
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9
views
gate2008
easy
linearalgebra
systemofequations
videosolution
0
votes
0
answers
GATE201424 Video Solution
If the matrix $A$ is such that $A= \begin{bmatrix} 2\\ −4\\7\end{bmatrix}\begin{bmatrix}1& 9& 5\end{bmatrix}$ then the determinant of $A$ is equal to ______.
asked
Apr 18, 2020
in
Linear Algebra
by
admin
(
573
points)

9
views
gate20142
linearalgebra
numericalanswers
easy
determinant
videosolution
0
votes
0
answers
GATE200623 Video Solution
$F$ 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
asked
Apr 18, 2020
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Linear Algebra
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admin
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573
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7
views
gate2006
linearalgebra
normal
matrices
videosolution
0
votes
0
answers
GATE199301.1 Video Solution
For the below question, one or more of the alternatives are correct. Write the code letter$(s)$ $a$, $b$, $c$, $d$ corresponding to the correct alternative$(s) $ in the answer book. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative ... $(0,0,\alpha)$ $(\alpha,0,0)$ $(0,0,1)$ $(0,\alpha,0)$
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Apr 18, 2020
in
Linear Algebra
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admin
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573
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16
views
gate1993
eigenvalue
linearalgebra
easy
videosolution
0
votes
0
answers
GATE201434 Video Solution
Which one of the following statements is TRUE about every $n \times n$ matrix with only real eigenvalues? If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative. ... eigenvalues are positive. If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.
asked
Apr 18, 2020
in
Linear Algebra
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admin
(
573
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17
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gate20143
linearalgebra
eigenvalue
normal
videosolution
0
votes
0
answers
GATE20199 Video Solution
Let $X$ be a square matrix. Consider the following two statements on $X$. $X$ is invertible Determinant of $X$ is nonzero 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
asked
Apr 18, 2020
in
Linear Algebra
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admin
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573
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8
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gate2019
engineeringmathematics
linearalgebra
determinant
videosolution
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