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
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.
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
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
x + 2y - 3z = a
2x + 3y + 3x = b
5x + 9y - 6z = c
This system is consistent if a, b and c satisfy the equation
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]
.
with n > 3 and aij = i.j. The rank of A is [2015 : 1 Mark, Set-II]
[2015 : 2 Marks, Set-I]
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