Linear Algebra

Question 31

Two eigenvalues of a 3 x 3 real matrix P are (2 + √ -1) and 3. The determinant of P is _____   

• 0

• 1

• 15

• -1

Question 32

Let f (x) be a polynomial and g(x) = f\'(x) be its derivative. If the degree of (f(x) + f(−x)) is 10, then the degree of (g(x) − g(−x)) is ___________ . [This Question was originally a Fill-in-the-blanks Question]

• 8
• 7
• 9
• 10

Question 33

Consider the systems, each consisting of m linear equations in n variables.

I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution

Which one of the following is CORRECT?

• I, II and III are true
• Only II and III are true
• Only III is true
• None of them is true

Question 34

Suppose that the eigenvalues of matrix A are 1, 2, 4. The determinant of (A⁻¹)^T is _________ [This Question was originally a Fill-in-the-blanks Question]

• 1/8
• 1
• 1/4
• 2

Question 35

How many antisymmetric relations are there on a set with n elements ?

• 2ⁿ.3^n(n-1)/2
• 2ⁿ
• n²
• n

Question 36

Consider the following set of equations

x+2y = 5
4x+8y = 12
3x+6y+3z = 15

This set-

• has a unique solution
• has no solutions
• has finite number of solutions
• has infinite number of solutions

Question 37

The rank of the matrix given below is:

1    4    8    7
0    0    3    0
4    2    3    1
3   12   24   21

• 3
• 1
• 2
• 4

Question 38

Consider the following determinant Which of the following is a factor of Δ ?

• a+b
• a-b
• a+b+c
• abc

Question 39

Derive the expression for the number of expressions required to solve a system of linear equations in n unknowns using the Gaussian Elimination Method. Assume that one operation refers to a multiplication followed by an addition.

Question 40

The determinant of the matrix is

• 11
• -48
• 0
• -24