Linear Algebra

Question 11

Consider the following system of equations in three real variables x₁, x₂ and x₃

2x1 - x2 + 3x3 = 1 
3x1- 2x2 + 5x3 = 2
-x1 + 4x2 + x3 = 3 

This system of equations has

• no solution

• a unique solution

• more than one but a finite number of solutions

• an infinite number of solutions

Question 12

What are the eigenvalues of the following 2 × 2 matrix? 

• -1 and 1

• 1 and 6

• 2 and 5

• 4 and -1

Question 13

The number of different n × n symmetric matrices with each element being either 0 or 1 is: (Note: power(2, x) is same as 2^x)

• power(2, n)
• power(2, n²)
• power(2, (n² + n)/2)
• power(2, (n² - n)/2)

Question 14

Let A, B, C, D be n × n matrices, each with non-­zero determinant. If ABCD = 1, then B^-1 is

• D^-1C^-1A^-1
  

• CDA
  

• ADC
  

• Does not necessarily exist
  

Question 15

An orthogonal matrix A has eigen values 1, 2 and 4. What is the trace of the matrix A^T?

• 0.125

• 1.75

• 7

• None of the above

Question 16

How many solutions does the following system of linear equations have ? 

  -x + 5y = -1
   x - y = 2
   x + 3y = 3

• unique
  

• two distinct solutions
  

• infinitely many
  

• none of these
  

Question 17

In an M\'N matrix such that all non-zero entries are covered in a rows and b columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is

• ≤ a + b
  

• ≤ max {a, b}
  

• ≤ min {M-a, N-b}
  

• ≤ min {a, b}
  

Question 18

Consider the following system of linear equations

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of a, does this system of equations have infinitely many solutions?

• 0
• 1
• 2
• infinitely many

Question 19

The rank of the given matrix is -

[caption width="800"] [/caption]

• 1
  

• 0
  

• 2
  

• 4
  

Question 20

Consider the following statements:

S1: The sum of two singular n × n matrices may be non-singular
S2: The sum of two n × n non-singular matrices may be singular. 

Which of the following statements is correct?

• S1 and S2 are both true
• S1 is true, S2 is false
• S1 is false, S2 is true
• S1 and S2 are both false