# 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.

### 8 thoughts on “Linear Algebra”

• 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