Linear Algebra

Question 1
If the sum and product of eigenvalues of a 2 \times 2 real matrix \begin{bmatrix} 3 & p\\ q & p \end{bmatrix} are 4 and -1 respectively, then |p| is ______ (in integer).
A
4
B
2
C
6
D
8
GATE ME 2022 SET-2   Engineering Mathematics
Question 1 Explanation: 
From the property of eigen values,
Sum of eigen values = Trace of matrix
4=3+q
q=1

Product of eigen values = Determinant
\begin{aligned} -1&=\begin{vmatrix} 3 &p \\ p& q \end{vmatrix}\\ 3q-p^2&=-1\\ 3-p^2&=-1\\ p^2&=4\\ p&=\pm 2\\ \Rightarrow |p|&=2 \end{aligned}
Question 2
A is a 3\times 5 real matrix of rank 2. For the set of homogeneous equations Ax = 0, where 0 is a zero vector and x is a vector of unknown variables, which of the following is/are true?
A
The given set of equations will have a unique solution.
B
The given set of equations will be satisfied by a zero vector of appropriate size.
C
The given set of equations will have infinitely many solutions.
D
The given set of equations will have many but a finite number of solutions.
GATE ME 2022 SET-2   Engineering Mathematics
Question 2 Explanation: 
Zero solution is always a solution of Ax = 0.
Option (b) is correct.
Given A is 3x5 real matrix and r(A) = 2 and Ax = 0 is a system of homogeneous linear equations since r(A) \lt number of unknown the system has infinite solution option (c) is also correct.
Question 3
The system of linear equations in real (x, y) given by
(x\;\;y)\begin{bmatrix} 2 & 5-2\alpha \\ \alpha & 1 \end{bmatrix} =(0\;\;0)
involves a real parameter \alpha and has infinitely many non-trivial solutions for special value(s) of \alpha . Which one or more among the following options is/ are non-trivial solution(s) of (x,y) for such special value(s) of \alpha ?
A
x=2,y=-2
B
x=-1,y=4
C
x=1,y=1
D
x=4,y=-2
GATE ME 2022 SET-1   Engineering Mathematics
Question 3 Explanation: 
\begin{aligned} \begin{pmatrix} x &y \end{pmatrix}\begin{pmatrix} 2 &5-2\alpha \\ \alpha & 1 \end{pmatrix}&=\begin{pmatrix} 0 &0 \end{pmatrix}\\ 2x+\alpha y&=0\\ (5-2\alpha )x+y&=0\\ \therefore \begin{pmatrix} 2 & \alpha \\ 5-2\alpha & 1 \end{pmatrix}\begin{pmatrix} x\\y \end{pmatrix}&=\begin{pmatrix} 0\\0 \end{pmatrix} \;\;\; . . . . (1) \end{aligned}
To get infinite number of non-trivial solutions \begin{aligned} \begin{vmatrix} 2 &\alpha \\ 5-2\alpha & 1 \end{vmatrix}&=0\\ 2-(5\alpha -2\alpha ^2)&=0\\ (2\alpha -1)(\alpha -2)&=0\\ \therefore \alpha =\frac{1}{2},\alpha &=2\\ \end{aligned}
At \alpha =\frac{1}{2} ; eq. (1) gives (4x+y)=0 . . .(2)
option (B) is satisfied by (2)
At \alpha =2 ; eq. (1) gives (x+y)=0 . . .(3)
option (A) is satisfied by (3)
Both options (A) and (B) are correct
Question 4
If A=\begin{bmatrix} 10 &2k+5 \\ 3k-3 & k+5 \end{bmatrix} is a symmetric matrix, the value of k is ___________.
A
8
B
5
C
-0.4
D
\frac{1+\sqrt{1561}}{12}
GATE ME 2022 SET-1   Engineering Mathematics
Question 4 Explanation: 
A=\begin{bmatrix} 10 & 2k+5\\ 3k-3 & k+5 \end{bmatrix}
(2k + 5) = (3k - 3)
k=8
Question 5
Consider an n x n matrix A and a non-zero n x 1 vector p. Their product Ap=\alpha ^2p, where \alpha \in \mathbb{R} and \alpha \notin \{-1,0,1\}. Based on the given information, the eigen value of A^2 is:
A
\alpha
B
\alpha ^2
C
\sqrt{\alpha }
D
\alpha ^4
GATE ME 2021 SET-2   Engineering Mathematics
Question 5 Explanation: 
Given, A P=\alpha^{2} P
By comparison with A X=\lambda X \Rightarrow
\Rightarrow \quad \lambda=\alpha^{2}
Hence, eigen value of A is \alpha^{2}, so eigen value of A^{2} is \alpha^{4}.
Question 6
Consider a vector p in 2-dimensional space. Let its direction (counter- clockwise angle with the positive x-axis) be \theta. Let p be an eigenvector of a 2 \times 2 matrix A with corresponding eigenvalue \lambda, \; \lambda, > 0. If we denote the magnitude of a vector v by ||v||, identify the VALID statement regarding p', where p'=Ap.
A
Direction of p'=\lambda \theta ,\; ||p'||=||p||
B
Direction of p'= \theta ,\; ||p'||=\lambda||p||
C
Direction of p'=\lambda \theta ,\; ||p'||=\lambda||p||
D
Direction of p'=\theta ,\; ||p'||=||p|| / \lambda
GATE ME 2021 SET-1   Engineering Mathematics
Question 6 Explanation: 
\because A is a 2 \times 2 matrix and P is the eigen vector of matrix A with corresponding eigen value \lambda
\begin{aligned} \text{Given}:\quad P^{\prime} &=P \\ A P &=\lambda P \\ P^{\prime} &=\lambda P\\ \text{Hence},\quad \left\|P^{\prime}\right\|&=\|\lambda P\|=\lambda|P|\\ \end{aligned}
But direction of vector P^{\prime} will be same as vector P.
Question 7
Let I be a 100 dimensional identity matrix and E be the set of its distinct (no value appears more than once in E ) real eigenvalues. The number of elements in E is ______.
A
1
B
100
C
10
D
0
GATE ME 2020 SET-2   Engineering Mathematics
Question 7 Explanation: 
I_{100}
Eigen values of I \rightarrow \underset{100 \text { times }}{\underbrace{1,1, \ldots \ldots \ldots ., 1}}
Set of distributed eigen value E=\{1\}
Number of elements in E=1
Question 8
A matrix P is decomposed into its symmetric part S and skew symmetric part V. If
S=\begin{pmatrix} -4 &4 &2 \\ 4& 3 & 7/2\\ 2& 7/2 & 2 \end{pmatrix}, V=\begin{pmatrix} 0 &-2 &3 \\ 2& 0 & 7/2\\ -3& -7/2 & 0 \end{pmatrix},
Then matrix P is
A
A
B
B
C
C
D
D
GATE ME 2020 SET-2   Engineering Mathematics
Question 8 Explanation: 
\begin{array}{l} S=\frac{P+P^{T}}{2} \\ V=\frac{P-P^{T}}{2} \\ P=S+V=\left(\begin{array}{ccc} -4 & 2 & 5 \\ 6 & 3 & 7 \\ -1 & 0 & 2 \end{array}\right) \end{array}
Question 9
Multiplication of real valued square matrices of same dimension is
A
associative
B
commutative
C
always positive definite
D
not always possible to compute
GATE ME 2020 SET-1   Engineering Mathematics
Question 9 Explanation: 
Matrix multiplication is associative.
Question 10
The transformation matrix for mirroring a point in x-y plane about the line y=x is given by
A
\begin{bmatrix} 1 & 0\\ 0&-1 \end{bmatrix}
B
\begin{bmatrix} -1 & 0\\ 0&1 \end{bmatrix}
C
\begin{bmatrix} 0 & 1\\ 1&0 \end{bmatrix}
D
\begin{bmatrix} 0 & -1\\ -1&0 \end{bmatrix}
GATE ME 2019 SET-2   Engineering Mathematics
Question 10 Explanation: 
The transformation matrix for mirroring a point in x-y plane about the line y = x is
\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]
There are 10 questions to complete.

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