# Linear Algebra

 Question 1
The smallest eigenvalue and the corresponding eigenvector of the matrix $\left[\begin{array}{cc} 2 & -2 \\ -1 & 6 \end{array}\right]$, respectively, are
 A 1.55 and $\left\{\begin{array}{l} 2.00 \\ 0.45 \end{array}\right\}$ B 2.00 and $\left\{\begin{array}{l} 1.00 \\ 1.00 \end{array}\right\}$ C 1.55 and $\left\{\begin{array}{l} -2.55 \\ -0.45 \end{array}\right\}$ D 1.55 and $\left\{\begin{array}{c} 2.00 \\ -0.45 \end{array}\right\}$
GATE CE 2021 SET-2   Engineering Mathematics
Question 1 Explanation:
\begin{aligned} A&=\left[\begin{array}{cc} 2 & -2 \\ -1 & 6 \end{array}\right] \Rightarrow|A-\lambda I|=0 \\ \Rightarrow \qquad \lambda&=(4+\sqrt{6}) \text { and }(4-\sqrt{6})\\ A X&=\lambda X\\ (A-\lambda I) X&=0 \end{aligned}
${\left[\begin{array}{cc} 2-(4-\sqrt{6}) & -2 \\ -1 & 6-(4-\sqrt{6}) \end{array}\right] \left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] =\left[\begin{array}{l} 0 \\ 0 \end{array}\right]}$
\begin{aligned} x_{1}&=\left(\frac{2}{-2+\sqrt{6}}\right) x_{2}\\ \text { Let, } \qquad x_{2}&=K \text { then } x_{1}=\left(\frac{2}{-2+\sqrt{6}}\right) \text { K }\\ \Rightarrow \qquad\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]&=\left[\begin{array}{c} \frac{2}{-2+\sqrt{6}} k \\ k \end{array}\right] \approx\left[\begin{array}{c} 2 \\ -2+\sqrt{6} \end{array}\right]=\left[\begin{array}{l} 2.00 \\ 0.45 \end{array}\right] \end{aligned}
 Question 2
Suppose that P is a 4x5 matrix such that every solution of the equation Px=0 is a scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T$. The rank of P is _______
 A 1 B 2 C 3 D 4
GATE CSE 2021 SET-2   Engineering Mathematics
Question 2 Explanation:
 Question 3
If A is a square matrix then orthogonality property mandates
 A $A A^{T}=I$ B $A A^{T}=0$ C $A A^{T}=A^{-1}$ D $A A^{T}=A^{2}$
GATE CE 2021 SET-2   Engineering Mathematics
Question 3 Explanation:
$\text { If, } \qquad \qquad A A^{\top}=I \quad \text { or } A^{-1}=A^{T}$
The matrix is orthogonal.
 Question 4
The rank of the matrix $\left[\begin{array}{cccc} 5 & 0 & -5 & 0 \\ 0 & 2 & 0 & 1 \\ -5 & 0 & 5 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right]$ is
 A 1 B 2 C 3 D 4
GATE CE 2021 SET-2   Engineering Mathematics
Question 4 Explanation:
\begin{aligned} \left[\begin{array}{cccc} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ -5 & 0 & -1 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right] & \stackrel{R_{1} \longleftrightarrow R_{1}+R_{3}}{\longrightarrow}\left[\begin{array}{llll} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \end{array}\right] \\ & \stackrel{R_{4} \longleftrightarrow R_{4}-\frac{1}{2} R_{2}}{\longrightarrow}\left[\begin{array}{llll} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{2} \end{array}\right]\\ &R_{3} \longleftrightarrow R_{4}\left[\begin{array}{llll}5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & \frac{3}{2} \\ 0 & 0 & 0 & 0\end{array}\right] \end{aligned}
Rank(A) = 3
 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 the following matrix.
$\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}$
The largest eigenvalue of the above matrix is __________.
 A 1 B 3 C 4 D 6
GATE CSE 2021 SET-1   Engineering Mathematics
Question 6 Explanation:
 Question 7
Let A be a $10\times10$ matrix such that $A^{5}$ is a null matrix, and let I be the $10\times10$ identity matrix. The determinant of $\text{A+I}$ is ___________________.
 A 1 B 2 C 4 D 8
GATE EE 2021   Engineering Mathematics
Question 7 Explanation:
\begin{aligned} \text{Given}:\qquad A^{5}&=0\\ A x &=\lambda x \\ \Rightarrow \qquad \qquad A^{5} x &=\lambda^{5} x \quad(\because x \neq 0) \\ \Rightarrow \qquad \qquad \lambda^{5} &=0 \\ \Rightarrow \qquad \qquad \lambda &=0 \end{aligned}
Eigen values of $A+I$ given $\lambda+1$
$\because$ Eigen values of $I_{A}=1$
Hence $|A+I|=$ Product of eigen values $=1 \times 1 \times 1 \times \ldots 10$times
$=1$
 Question 8
A real $2\times2$ non-singular matrix A with repeated eigenvalue is given as
$A=\begin{bmatrix} x & -3.0\\ 3.0 & 4.0 \end{bmatrix}$
where x is a real positive number. The value of x (rounded off to one decimal place) is ________________
 A 5.2 B 18.7 C 10 D 6.8
GATE EC 2021   Engineering Mathematics
Question 8 Explanation:
$A=\left[\begin{array}{cc} x & -3 \\ 3 & 4 \end{array} \right]$
Characteristic equation
\begin{aligned} |A-\lambda I| &=\left|\begin{array}{cc} x-\lambda & -3 \\ 3 & 4-\lambda \end{array}\right|=0 \\ \Rightarrow \quad \lambda^{2}-(4+x) \lambda+(4 x+9)&=0 \end{aligned}
Roots are repeating
\begin{aligned} \Rightarrow \qquad b^{2}-4 a c&=0\\ \Rightarrow \qquad \left(4+x^{2}\right)-4(4 x+9) &=0 \\ 16+x^{2}+8 x-16 x-20 &=0 \\ x &=10 \\ \Rightarrow \qquad x^{2}-8 x+20 &=0\\ x&=\frac{8 \pm \sqrt{64+80}}{2} \\ &=\frac{8 \pm 12}{2}=-2,10 \end{aligned}
Since, x is positive
$\therefore x=10$
 Question 9
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 9 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 10
If the vectors $(1.0,\:-1.0,\:2.0), (7.0,\:3.0,\:x)$ and $(2.0,\:3.0,\:1.0)$ in $\mathbb{R}^{3}$ are linearly dependent, the value of x is __________
 A 8 B 9 C 4 D 2
GATE EC 2021   Engineering Mathematics
Question 10 Explanation:
$(1,-1,2)$
$(7,3, x)$ are linearly dependent when x=?
$(2,3,1)$
\begin{aligned} \Rightarrow\left|\begin{array}{ccc} 1 & -1 & 2 \\ 7 & 3 & x \\ 2 & 3 & 1 \end{array}\right| &=0 \Rightarrow 1(3-3 x)+1(7-2 x)+2(15)=0 \\ \Rightarrow \quad-5 x&=-40 \\ x&=8 \end{aligned}

There are 10 questions to complete.

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