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

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
Answer: (b)
Solution: Volume of water


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




Answer: (d)

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

∴ Complex Eigen values and complex Eigen vectors.

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




Answer: (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 6: For the given orthogonal matrix Q,    [2018 : 1 Marks, Set-I]

The inverse is

Answer: (c)
Solution:



Or ∵ Q is orthogonal
∴ Q^-1 = Q^T 

Question 7:  is equal to    [2017 : 2 Marks, Set-II]

Answer: (a)
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 
Answer: (a)
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.
Answer: (a)
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
Answer: (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 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
Answer: (b)

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
Answer: (d)
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
Answer: (d)
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 a_ij = i.j. The rank of A is    [2015 : 1 Mark, Set-II]
(a) 0 
(b) 1 
(c) n - 1 
(d) n
Answer: (b)
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
Answer: (d)
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