Classical

Linear Algebra - Linear Algebra

1:  

Rank of the matrix A =

 
0 0 0 0
4 2 3 0
1 0 0 0
4 0 3 0
 
A.

0

B.

1

C.

2

D.

3

 
 

Option: D

Explanation :

Determinant A = 0

4 2 3
1 0 0
4 0 3

Hence rank of A=3


2:  

A set of linear equations is represented by the matrix equation Ax = b. The necessary condition for the existence of a solution for this system is

A. A must be invertible
B. b must be linearly depended on the columns of A
C. b must be linearly independent of the columns of A
D. None of these
 
 

Option: B

Explanation :


3:  

Consider the following two statements:
I. The maximum number of linearly independent column vectors of a matrix A is called the rank of A.
II. If A is an n x n square matrix, it will be nonsingular is rank A = n.
With reference to the above statements, which of the following applies?

A. Both the statements are false
B. Both the statements are true
C. I is true but II is false.
D. I is false but II is true.
 
 

Option: B

Explanation :


4:  

The system of linear equations
(4d - 1)x +y + z = 0
- y + z = 0
(4d - 1) z = 0
has a non-trivial solution, if d equals

A.

1/2

B.

1/4

C.

3/4

D.

1

 
 

Option: B

Explanation :

The system of homogeneous linear equations has a non-trivial solution if

 
4d-1 1 1
0 -1 1
0 0 4d-1
 
= 0

=> -(4d-1)2 = 0

=> d = 1/4


5:  

The rank of a 3 x 3 matrix C (= AB), found by multiplying a non-zero column matrix A of size 3 x 1 and a non-zero row matrix B of size 1 x 3, is

A.

0

B.

1

C.

2

D.

3

 
 

Option: A

Explanation :

Let A =
 
a1
b1
c1
 
; B =
 
a2 b2 c2
 
;

C =
 
a1a2   a1b2   a1c2
b1a2   b1b2   b1c2
c1a1   c1b2   c1c2
 




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