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 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 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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