# Linear Algebra

 Question 1
If the sum and product of eigenvalues of a $2 \times 2$ real matrix $\begin{bmatrix} 3 & p\\ p & q \end{bmatrix}$ 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.

### 10 thoughts on “Linear Algebra”

1. • Dear Sangram Kumar
Thank you for your suggestions. We have updated the correction suggested by You.

2. Great job sir 👍

3. In question a12 should be -i

• In question 24

4. The answer for question no 4 should be A and it is marked as B

• 5. Sir in question no 24 first row second couloum should be -i

6. 7. 