Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev

Topic wise GATE Past Year Papers for Civil Engineering

GATE : Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev

The document Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev is a part of the GATE Course Topic wise GATE Past Year Papers for Civil Engineering.
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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 (Part - 1) Civil Engineering (CE) Notes | EduRev
(a) 156π
(b) 396π
(c) 468π 
(d) 78π

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

Q.4. The matrix Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev 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 (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev

Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
 Complex Eigenvalues and complex Eigen vectors.

Q.5. Which one of the following matrices is singular?    
[2018: 1 Marks, Set-I]
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
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.

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 (Part - 1) Civil Engineering (CE) Notes | EduRevand B = Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev ABT is equal to    
[2017 : 2 Marks, Set-II]
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Ans. (a)
Solution. 
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev

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 (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
|A - λI| = 0
(3 - λ) (1 - λ) - 8 = 0
3 - 4λ + λ2 - 8 = 0
λ2 - 4λ - 5 = 0

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

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

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

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

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

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 (Part - 1) Civil Engineering (CE) Notes | EduRev is _____.    [2014 : 2 Marks, Set-II]
Solution:
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Determinant of matrix Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev is not zero.
∴ Rank is 2
Question 18: The determinant of matrix Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev    [2014 : 1 Mark, Set-II]
Solution: 
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Interchanging column 1 and column 2 and taking transpose,
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev

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 (Part - 1) Civil Engineering (CE) Notes | EduRev
Area of triangle is
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev

Question 20: The sum of Eigen values of matrix, [M] is where Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev     [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 (Part - 1) Civil Engineering (CE) Notes | EduRev Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev  the product KT JK is ____.    [2014 : 1 Mark, Set-I]
Solution:
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev

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 (Part - 1) Civil Engineering (CE) Notes | EduRev   [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 (Part - 1) Civil Engineering (CE) Notes | EduRev
The characteristic equation is,
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
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 (Part - 1) Civil Engineering (CE) Notes | EduRev    [2010 : 2 Marks]
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Answer: (b)
Solution:
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev
Linear Algebra (Part - 1) Civil Engineering (CE) Notes | EduRev

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