Linear Algebra

Question 1
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 1 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 2
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 2 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 3
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 3 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 4
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 4 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 5
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 5 Explanation: 
Matrix multiplication is associative.
Question 6
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 6 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]
Question 7
In matrix equation [A]{X}={R},
[A]=\begin{bmatrix} 4 & 8 & 4\\ 8& 16 & -4\\ 4& -4 & 15 \end{bmatrix}, \{X\}=\begin{Bmatrix} 2\\ 1\\ 4 \end{Bmatrix} and \{R\}=\begin{Bmatrix} 32\\ 16\\ 64 \end{Bmatrix}.
One of the eigenvalues of matrix [A] is
A
4
B
8
C
15
D
16
GATE ME 2019 SET-2   Engineering Mathematics
Question 7 Explanation: 
Given that AX=R
\begin{array}{l} \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=\left[\begin{array}{l} 32 \\ 16 \\ 64 \end{array}\right] \\ \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=16\left[\begin{array}{c} 32 \\ 4 \end{array}\right](\because A X=\lambda X) \end{array}
\therefore One of eigen value of the given matrix A is given by \lambda=16
Question 8
The set of equations
x+y+z=1 \\ax-ay+3z=5 \\5x-3y+az=6
has infinite solutions, if a=
A
-3
B
3
C
4
D
-4
GATE ME 2019 SET-1   Engineering Mathematics
Question 8 Explanation: 
The given system of equations can be expressed in the matrix form:
\left[\begin{array}{ccc} 1 & 1 & 1 \\ a & -a & 3 \\ 5 & -3 & a \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 1 \\ 5 \\ 6 \end{array}\right]
\mathrm{AX}=\mathrm{B} form
\text { Where } A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ a & -a & 3 \\ 5 & -3 & a \end{array}\right] X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \& B=\left[\begin{array}{c} 1 \\ 5 \\ 6 \end{array}\right]
The augmented matrix is
[A / B]=\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1 \\ a & -a & 3 & 5 \\ 5 & -3 & a & 6 \end{array}\right]
\mathrm{R}_{2}-\mathrm{aR}_{1} \& \mathrm{R}_{3}-5 \mathrm{R}_{1}
[\mathrm{A} / \mathrm{B}]=\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1 \\ 0 & -2 \mathrm{a} & 3-\mathrm{a} & 5-\mathrm{a} \\ 0 & -8 & \mathrm{a}-5 & 1 \end{array}\right]
\mathrm{aR}_{3}-4 \mathrm{R}_{2}
[\mathrm{A} / \mathrm{B}]=\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1 \\ 0 & -2 \mathrm{a} & 3-\mathrm{a} & 5-\mathrm{a} \\ 0 & 0 & \mathrm{a}^{2}-\mathrm{a}-12 & 5 \mathrm{a}-20 \end{array}\right]
For infinite solutions
\rho(\mathrm{A} / \mathrm{B})=\rho(\mathrm{A}) \lt Number of unknowns
\Rightarrow a^{2}-a-12=0 \& 5 a-20=0
\Rightarrow a=4
Question 9
Consider the matrix
P=\begin{bmatrix} 1 & 1 &0 \\ 0&1 &1 \\ 0& 0 & 1 \end{bmatrix}
The number of distinct eigenvalues of P is
A
0
B
1
C
2
D
3
GATE ME 2019 SET-1   Engineering Mathematics
Question 9 Explanation: 
\text { Given: } A=\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]
It is an upper triangular matrix. It's diagonal elements are eigen values.
The eigen values of the matrix are 1, 1, 1.
\therefore Number of distinct eigen values = 1
Hence, option (B) is correct.
Question 10
The problem of maximizing z=x_{1}-x_{2} subject to constraints x_{1}+x_{2}\leq 10, x_{1}\geq 0,x_{2}\geq 0 and x_{2}\leq 5 has
A
no solution
B
one solution
C
two solutions
D
more than two solutions
GATE ME 2018 SET-2   Engineering Mathematics
Question 10 Explanation: 


\begin{aligned} \text{Max} ., Z=x_{1}-x_{2}\\ Constratins : x_{1}+x_{2} \leq 10 \\ x_{1} \geq 0 ; x_{2} \geq 0 & \text { and } x_{2} \leq 5 \\ Z(0,5)=0-5 &=-5 \\ Z(5,5)=5-5=0 & \\ Z(10,0)=10-0=& 10 \\ Z_{\max }=10 \text { at }(10,0) \end{aligned}
\therefore The problem has one solution
There are 10 questions to complete.

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