Linear Algebra

Question 21

The determinant of the matrix is 

• 4
  

• 20
  

• 0
  

• 5
  

Question 22

In the LU decomposition of the matrix

| 2  2 |
| 4  9 |

, if the diagonal elements of U are both 1, then the lower diagonal entry l₂₂ of L is

• 4
• 5
• 6
• 7

Question 23

Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are –1 and 7. What are the values of a and b? 

A = | 1   4  |
      |  b  a  |

• a = 3, b = 5 
  

• a = 5, b = 3
  

• a = 4, b = 6
  

• a = 6, b = 4 
  

Question 24

The larger of the two eigenvalues of the matrix

⎡ 4   5 ⎤
⎣ 2   1 ⎦

is ____

• 5
• 6
• 7
• 8

Question 25

Perform the following operations on the matrix 

⎡ 3  4   45⎤
⎢ 7  8  105⎥
⎣13  2  195⎦ 

1. Add the third row to the second row.
2. Subtract the third column from the first column.

The determinant of the resultant matrix is _____________.

• 50
  

• 1
  

• 0
  

• 100
  

Question 26

In the given matrix, one of the eigenvalues is 1. the eigenvectors corresponding to the eigenvalue 1 are

⎡ 1 -1  2 ⎤
⎢ 0  1  0 ⎥
⎣ 1  2  1 ⎦

• A
• B
• C
• D

Question 27

If the following system has non-trivial solution,

  px + qy + rz = 0
  qx + ry + pz = 0
  rx + py + qz = 0 

then which one of the following options is True?

• p – q + r = 0 or p = q = –r
• p + q – r = 0 or p = –q = r
• p + q + r = 0 or p = q = r
• p – q + r = 0 or p = –q = –r

Question 28

Let a(x, y), b(x, y,) and c(x, y) be three statements with variables x and y chosen from some universe. Consider the following statement:

(∃x)(∀y)[(a(x, y) ∧ b(x, y)) ∧ ¬c(x, y)] Which one of the following is its equivalent?  

• (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]
• (∃x)(∀y)[(a(x, y) ∨ b(x, y)) ∧¬ c(x, y)]
• ¬ (∀x)(∃y)[(a(x, y) ∧ b(x, y)) → c(x, y)]
• ¬ (∀x)(∃y)[(a(x, y) ∨ b(x, y)) → c(x, y)]

Question 29

What values of x, y and z satisfy the following system of linear equations? 

[caption width="800"] [/caption]

• x=6, y=3, z=2

• x=12, y=3, z=-4

• x=6, y=6, z=-4

• x=12, y=-3, z=0

Question 30

Let H1,H2,H3,... be harmonic numbers Then, for   can be expressed as

• nHn+1 - (n + 1)
• (n + 1)Hn - n
• nHn - n
• (n+1)Hn+1—(n+1)