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.

Try yourself:For the given orthogonal matrix Q,    [2018: 1 Marks, Set-I]

The inverse is

• A.

  
• B.

  
• C.

  
• D.

  

Explanation

Or ∵ Q is orthogonal.
∴ Q^-1 = Q^T 

View Solution

Q.7. If  A = and B =  AB^T 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

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]

• 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 eigenvectors exists.
• C.

  Eigenvalue 3 has a multiplicity of 2, and no independent eigenvector exists.
• D.

  Eigenvalue are 3 and -3, and two independent eigenvectors exist.

Explanation

► 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.

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: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]

• A.

  -2
• B.

  1
• C.

  7 / 3
• D.

  14 / 3

Explanation

Let λ₁ and λ₂ be the eigenvalue 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 the ratio of two eigenvalues 

View Solution

Try yourself:Let A = 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

Explanation

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.

View Solution

Try yourself: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

Explanation

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

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  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
Answer: (a)
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
Answer: (a)
Solution: Sum of eigen values = trace of matrix
= 215 + 150 + 550 = 915

Question 21: Given the matrices    the product K^T 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,
Q_2x4 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
Answer: (b)
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
Answer: (d)
Solution: Since (A + A^t) = A^t + (A^t)^t 
= A^t + A
i.e. S^t = S
∴ S is symmetric
Since (A - A^t)^t = A^t - (A^t)^t 
= A^t - A = -(A - A^t)
i.e. D^t = - D
So D is Skew-Symmetric.

Question 25: The inverse of the matrix     [2010 : 2 Marks]

Answer: (b)
Solution: