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

Document Description: Linear Algebra for Civil Engineering (CE) 2022 is part of Topic wise GATE Past Year Papers for Civil Engineering preparation. The notes and questions for Linear Algebra have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Linear Algebra covers topics like and Linear Algebra Example, for Civil Engineering (CE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Linear Algebra.

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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 Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Q.2. The inverse of the matrix Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)    
[2019: 2 Marks, Set-ll]
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study 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 Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study 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 Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)

Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)and B = Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) ABT is equal to    
[2017 : 2 Marks, Set-II]
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Ans. (a)
Solution. 
Linear Algebra Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) is _____.    [2014 : 2 Marks, Set-II]
Solution:
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Determinant of matrix Linear Algebra Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)    [2014 : 1 Mark, Set-II]
Solution: 
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Interchanging column 1 and column 2 and taking transpose,
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Area of triangle is
Linear Algebra Notes | Study 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 Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) Linear Algebra Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study 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 Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
The characteristic equation is,
Linear Algebra Notes | Study 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 Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)    [2010 : 2 Marks]
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Answer: (b)
Solution:
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)

The document Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Topic wise GATE Past Year Papers for Civil Engineering.
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