Rank of the matrix A =

A.  0 

B.  1 

C.  2 

D.  3 

Option: D Explanation :
Determinant A = 0
Hence rank of A=3 
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 : 
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 : 
The system of linear equations
(4d  1)x +y + z = 0
 y + z = 0
(4d  1) z = 0
has a nontrivial 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 nontrivial solution if
=> (4d1)^{2} = 0 => d = 1/4 