Linear Algebra

# 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.

Introduction of Linear Algebra in English is available as part of our Topic wise GATE Past Year Papers for Civil Engineering for Civil Engineering (CE) & Linear Algebra in Hindi for Topic wise GATE Past Year Papers for Civil Engineering course. Download more important topics related with notes, lectures and mock test series for Civil Engineering (CE) Exam by signing up for free. Civil Engineering (CE): Linear Algebra Notes | Study Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
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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]

(a) 156π
(b) 396π
(c) 468π
(d) 78π

Ans. (b)
Solution. Volume of water

Q.2. The inverse of the matrix
[2019: 2 Marks, Set-ll]

Ans. (d)
Q.3. The rank of the following matrix is
[2018 : 2 Marks, Set-II]
(a) 1
(b) 2
(c) 3
(d) 4
Ans. (b)
Solution.

► Number of non zero rows = 2
► Rank of A = 2

Q.4. The matrix  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.

Complex Eigenvalues and complex Eigen vectors.

Q.5. Which one of the following matrices is singular?
[2018: 1 Marks, Set-I]

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]

The inverse is

Q.7. If  A = and B =  ABT is equal to
[2017 : 2 Marks, Set-II]

Ans. (a)
Solution.

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.

|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  Which one of the following statements is TRUE for the eigenvalues and eigenvectors of this matrix?    [2017: 2 Marks, Set-I]

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

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]

Question for Linear Algebra
Try yourself:The two Eigenvalues of the matrix  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]

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

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

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  is _____.    [2014 : 2 Marks, Set-II]
Solution:

Determinant of matrix  is not zero.
∴ Rank is 2
Question 18: The determinant of matrix     [2014 : 1 Mark, Set-II]
Solution:

Interchanging column 1 and column 2 and taking transpose,

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
Solution:

Area of triangle is

Question 20: The sum of Eigen values of matrix, [M] is where      [2014 : 1 Mark, Set-I]
(a) 915
(b) 1355
(c) 1640
(d) 2180
Solution: Sum of eigen values = trace of matrix
= 215 + 150 + 550 = 915

Question 21: Given the matrices    the product KT JK is ____.    [2014 : 1 Mark, Set-I]
Solution:

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    [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
Solution: We need eigen values of
The characteristic equation is,

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
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     [2010 : 2 Marks]

Solution:

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