Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) PDF Download

Q.1. Consider the hemispherical tank of radius 13 m as shown in the figure (not drawn to scale). What is the volume of water (in m3) when the depth of water at the centre of the tank is 6 m?    
[2019: 2 Marks, Set-ll]
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
(a) 156π
(b) 396π
(c) 468π 
(d) 78π

Ans. (b)
Solution. Volume of water
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Q.2. The inverse of the matrix Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)    
[2019: 2 Marks, Set-ll]
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Ans. (d)
Q.3. The rank of the following matrix is Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)    
[2018 : 2 Marks, Set-II]
(a) 1 
(b) 2
(c) 3 
(d) 4
Ans. (b)
Solution. 
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
► Number of non zero rows = 2
► Rank of A = 2

Q.4. The matrix Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) has    
[2018 : 2 Marks, Set-II]
(a) real eigenvalues and eigenvectors 
(b) real eigenvalues but complex eigenvectors 
(c) complex eigenvalues but real eigenvectors 
(d) complex eigenvalues and eigenvectors
Ans. (d)
Solution. 
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)

Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
 Complex Eigenvalues and complex Eigen vectors.

Q.5. Which one of the following matrices is singular?    
[2018: 1 Marks, Set-I]
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Ans. (c)
Solution. Option (a): |A| = 6 - 5 = 1
Option (b): |A| = 9 - 4 = 5
Option (c): |A| = 12-12 = 0
Option (d): |A| = 8 - 18 = -10
Hence matrix (c) is singular.

Question for Linear Algebra
Try yourself:For the given orthogonal matrix Q,    [2018: 1 Marks, Set-I]
Linear Algebra (Part - 1)
The inverse is
View Solution


Q.7. If  A = Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)and B = Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) ABT is equal to    
[2017 : 2 Marks, Set-II]
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Ans. (a)
Solution. 
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)

Q.8. Consider the following simultaneous equations (with c1 and c2 being constants):    
[2017 : 1 Mark, Set-II]
3x1 + 2x2 = c1 
4x1 + x2 = c2 
The characteristics equation for these simultaneous equations is 
(a) λ2 - 4λ - 5 = 0 
(b) λ2 - 4λ + 5 = 0 
(c) λ2 + 4λ - 5 = 0 
(d) λ2 + 4λ + 5 = 0 
Ans. (a)
Solution. 
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
|A - λI| = 0
(3 - λ) (1 - λ) - 8 = 0
3 - 4λ + λ2 - 8 = 0
λ2 - 4λ - 5 = 0

Question for Linear Algebra
Try yourself:Consider the matrix Linear Algebra (Part - 1) Which one of the following statements is TRUE for the eigenvalues and eigenvectors of this matrix?    [2017: 2 Marks, Set-I]
View Solution


Q.10. The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct?  [2017: 1 Mark, Set-I]
(a) PQ = I but QP ≠ I 
(b) QP = I but PQ ≠ I 
(c) PQ = I and QP = I 
(d) PQ = QP = I
Ans. (c)
Solution. Given that P is inverse of Q.
► P = Q-1   
► PQ = Q-1Q,  QP = QQ-1
► PQ = I , QP = I
∴ PQ = QP = I

Question for Linear Algebra
Try yourself:Consider the following linear system.    [2016 : 2 Marks, Set-Il]
x + 2y - 3z = a 
2x + 3y + 3x = b 
5x + 9y - 6z = c
This system is consistent if a, b and c satisfy the equation 
View Solution

Question for Linear Algebra
Try yourself:If the entries in each column of a square matrix M add up to 1, then an eigen value of M is [2016 : 1 Mark, Set - I]
View Solution

Question for Linear Algebra
Try yourself:The two Eigenvalues of the matrix Linear Algebra (Part - 1) have a ratio of 3 : 1 for p = 2. What is another value of p for which the Eigenvalues have the same ratio of 3 : 1?   
[2015: 2 Marks, Set-II]
View Solution

Question for Linear Algebra
Try yourself:Let A =Linear Algebra (Part - 1) with n > 3 and aij = i.j. The rank of A is    [2015 : 1 Mark, Set-II]
View Solution

Question for Linear Algebra
Try yourself:The smallest and largest Eigen values of the following matrix are Linear Algebra (Part - 1)    [2015 : 2 Marks, Set-I]
View Solution

Question 16: For what value of p the following set of equations will have no solution?    [2015 : 1 Mark, Set-I]
2x + 3y = 5
3x + py = 10
Solution: 
Given system of equations has no solution if the lines are parallel i.e., their slopes are equal
2/3 = 3/p
⇒ p = 4.5
Question 17: The rank of the matrix Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) is _____.    [2014 : 2 Marks, Set-II]
Solution:
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Determinant of matrix Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) is not zero.
∴ Rank is 2
Question 18: The determinant of matrix Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)    [2014 : 1 Mark, Set-II]
Solution: 
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Interchanging column 1 and column 2 and taking transpose,
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)

Question 19: With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have the following coordinates; (x1, y1) = (1, 0); (x2, y2) = (2, 2); (x3, y3) = (4, 3). The area of the triangle is equal to    [2014 : 1 Mark, Set-I]
(a) 3/2
(b) 3/4
(c) 4/5
(d) 5/2
Answer: (a)
Solution: 
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Area of triangle is
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)

Question 20: The sum of Eigen values of matrix, [M] is where Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)     [2014 : 1 Mark, Set-I]
(a) 915 
(b) 1355 
(c) 1640 
(d) 2180
Answer: (a)
Solution: Sum of eigen values = trace of matrix
= 215 + 150 + 550 = 915

Question 21: Given the matrices Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)  the product KT JK is ____.    [2014 : 1 Mark, Set-I]
Solution:
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)

Question 22: There are three matrixes P(4 x 2), Q(2 x 4) and R(4 x 1). The minimum of multiplication required to compute the matrix PQR is    [2013 : 1 Mark]
Solution: If we multiply QR first then,
Q2x4 x R(4x1) having multiplication number 8.
There fore P(4 x 2) QR(2 x 1) will have minimum number of multiplication = (8 + 8) = 16.

Question 23: The eigen values of matrix Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)   [2011 : 2 Marks] 
(a) -2.42 and 6.86 
(b) 3.48 and 13.53 
(c) 4.70 and 6.86 
(d) 6.86 and 9.50
Answer: (b)
Solution: We need eigen values of Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
The characteristic equation is,
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
So eigen values are,
λ = 3.48, 13.53

Question 24: [A] is square matix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] - [A]T, respectively. Which of the following statements is TRUE?    [2011 : 1 Mark]
(a) Both [S] and [D] are symmetric
(b) Both [S] and [D] are skew-symmetric 
(c) [S] is skew-symmetric and [D] is symmetric 
(d) [S] is symmetric and [D] is skew-symmetric
Answer: (d)
Solution: Since (A + At) = At + (At)t 
= At + A
i.e. St = S
∴ S is symmetric
Since (A - At)t = At - (At)t 
= At - A = -(A - At)
i.e. Dt = - D
So D is Skew-Symmetric.

Question 25: The inverse of the matrix Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)    [2010 : 2 Marks]
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Answer: (b)
Solution:
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)

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FAQs on Linear Algebra - Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)

1. What is linear algebra and why is it important for the GATE exam?
Ans. Linear algebra is a branch of mathematics that deals with vector spaces and linear equations. It is important for the GATE exam as it forms the foundation for various topics in engineering and computer science, such as machine learning, image processing, and control systems. Understanding linear algebra helps in solving complex problems and analyzing data efficiently.
2. How can I prepare for the linear algebra portion of the GATE exam?
Ans. To prepare for the linear algebra portion of the GATE exam, it is important to start by understanding the basic concepts such as vectors, matrices, and systems of linear equations. Practice solving numerical problems and solving exercises from textbooks or online resources. Make use of study materials, video lectures, and online tutorials to clarify any doubts. Additionally, solving previous years' GATE question papers will give you an idea of the exam pattern and the type of questions asked.
3. What are the key topics in linear algebra for the GATE exam?
Ans. The key topics in linear algebra for the GATE exam include vector spaces, linear transformations, eigenvalues and eigenvectors, diagonalization, inner product spaces, and orthogonal projections. It is important to have a strong understanding of these topics and their applications to solve problems in engineering and computer science.
4. Are there any specific tips to solve linear algebra questions efficiently in the GATE exam?
Ans. Yes, here are some tips to solve linear algebra questions efficiently in the GATE exam: 1. Understand the question thoroughly before attempting to solve it. 2. Break down complex problems into smaller steps and solve them one by one. 3. Use the properties and theorems of linear algebra to simplify calculations. 4. Practice mental calculations and use shortcuts when possible. 5. Keep track of time and allocate sufficient time for each question. 6. Solve previous years' GATE question papers to familiarize yourself with the exam pattern and improve your speed.
5. Can you suggest any additional resources to study linear algebra for the GATE exam?
Ans. Yes, apart from textbooks and study materials, there are several additional resources that can help you study linear algebra for the GATE exam. 1. Online video lectures and tutorials on platforms like YouTube and NPTEL. 2. Online courses specifically designed for GATE preparation, such as those offered by edX or Coursera. 3. Linear algebra textbooks authored by renowned mathematicians, such as Gilbert Strang's "Introduction to Linear Algebra" or Serge Lang's "Linear Algebra." 4. Online forums and discussion boards where you can interact with fellow GATE aspirants and clarify doubts.
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