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

## Topic wise GATE Past Year Papers for Civil Engineering

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## Civil Engineering (CE) : 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 Civil Engineering (CE) Course Topic wise GATE Past Year Papers for Civil Engineering.
All you need of Civil Engineering (CE) at this link: Civil Engineering (CE)

Question 1: Consider the hemi-spherical 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

Solution: Volume of water Question 2: The inverse of the matrix [2019 : 2 Marks, Set-ll]    Question 3: The rank of the following matrix is [2018 : 2 Marks, Set-II]
(a) 1
(b) 2
(c) 3
(d) 4
Solution:  Number of non zero rows = 2
rank of A = 2

Question 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
Solution:       Complex Eigen values and complex Eigen vectors.

Question 5: Which one of the following matrices is singular?    [2018 : 1 Marks, Set-I]    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 6: For the given orthogonal matrix Q,    [2018 : 1 Marks, Set-I] The inverse is Solution:   Or ∵ Q is orthogonal
∴ Q-1 = QT

Question 7: is equal to    [2017 : 2 Marks, Set-II] Solution: Question 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
Solution:  |A - λI| = 0
(3 - λ) (1 - λ) - 8 = 0
3 - 4λ + λ2 - 8 = 0
λ2 - 4λ - 5 = 0

Question 9: Consider the matrix Which one of the following statements is TRUE for the eigenvalues and eigen vectors of this matrix?    [2017 : 2 Marks, Set-I]
(a) Eigenvalue 3 has a multiplicity of 2, and only one independent eigenvector exists.
(b) Eigenvalue 3 has a multiplicity of 2, and two independent eigenvector exists.
(c) Eigenvalue 3 has a multiplicity of 2, and no independent eigenvector exists.
(d) Eigenvalue are 3 and -3, and two indepedent eigenvectors exist.
Solution: Characteristics equation is |A — λI| = 0 5 - 5λ, - λ, + λ2 + 4 = 0

λ2 - 6λ + 9 = 0
λ = 3, 3

Algebraic multiplicity of eigen value 3 is 2. It has only one independent eigen vector exists.

Question 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
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 11: 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
(a) 7 a - b - c = 0
(b) 3a + b - c = 0
(c) 3 a - b + c = 0
(d) 7 a - b + c = 0

Question 12: 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]
(a) 4
(b) 3
(c) 2
(d) 1
Solution: Consider the ‘2 x 2’ square matrix ⇒ λ2- (a + d)λ + (ad - be) = 0       .......(i)
Putting λ = 1, we get
1 - (a + d) + ad - bc = 0
1 - a - d + ad - (1 - d)(1 - a) = 0
1 - a - d + ad 1 + a+ d - ad = 0
0 = 0 which is true.

∴ λ = 1 satisfied the eq. (i) but λ = 2, 3, 4 does not satisfy the eq. (i). For all possible values of a, d.

Question 13: The two Eigen values of the matrix have a ratio of 3 : 1 for p = 2. What is another value of p for which the Eigen values have the same ratio of 3 : 1?    [2015 : 2 Marks, Set-II]
(a) -2
(b) 1
(c) 7/3
(d) 14/3
Solution: Let λ1 and λ2 be the eigen value of matrix A Sum of eigen value
= λ1 + λ2 = 2 + p     .......(i)
Product of eigen value
λ1λ2 = 2p - 1   ........(ii)      By putting values of p from options.
By putting option (d) 14/3 in above equations gives value Hence ratio of two eigen values Question 14: with n > 3 and aij = i.j. The rank of A is    [2015 : 1 Mark, Set-II]
(a) 0
(b) 1
(c) n - 1
(d) n
Solution: Rank of A = 1
Because each row will be scalar multiple of first row. So we will get only one non-zero row in row Echeleaon form of A
Alternative:
Rank of A = 1
Because all the minors of order greater than 1 will be zero.

Question 15: The smallest and largest Eigen values of the following matrix are [2015 : 2 Marks, Set-I]
(a) 1.5 and 2.5
(b) 0.5 and 2.5
(c) 1.0 and 3.0
(d) 1.0 and 2.0
Solution: For eigen values |A - λI| = 0 Only 1 and 2 satisfy this equation.

λ = 1, 1,2
Hence,Smallest eigen value = 1 and
Largest eigen value = 2

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