Linear Algebra (Part - 2) | Additional Documents & Tests for Civil Engineering (CE) PDF Download

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: