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

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Bharat said: (6:37am on Thursday 29th December 2016)
When we convert it in normal form we get rank=1

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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: A

Explanation :

x=A-1b

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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 :

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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

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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: B

Explanation :

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

C =
 
a1a2   a1b2   a1c2
b1a2   b1b2   b1c2
c1a1   c1b2   c1c2
 

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jaina nithin said: (7:42am on Thursday 13th February 2014)
there will be no zero rows in the resultant matrix then the rank should be 3
saurabh said: (7:37am on Wednesday 29th July 2015)
rank (AB) <= min { rank (A), rank (B)} so it should be 1
momit said: (10:24am on Tuesday 18th August 2015)
rankA=0 rankB=1rankAB_
Sherry said: (4:46am on Thursday 15th October 2015)
it should be 1..as rank of product of matrices do not excedd the minimum rank of two matrices
naveen said: (6:39pm on Saturday 20th August 2016)
it should be. determinant of 3*3 and 2*2 matrix is zero. so rank should be 1. only null matrix has 0 rank.
Amol said: (2:11am on Sunday 18th September 2016)
Rank shld be 3 , how come 0?
asit kr saha said: (9:26am on Tuesday 25th October 2016)
so whay the result of this matrix is equal to zero ??? i think its rank is 3
GAURAV SHARMA said: (3:36am on Saturday 29th October 2016)
|C|=0 so rank of C is <3 but there are no non-zero minors of order 2x2 so but there exists a 1x1 minor if any element in A and B is not zero then rank of C is 1
suraj said: (8:28am on Tuesday 24th January 2017)
rank should be 1
Rajnikant said: (10:09pm on Saturday 25th March 2017)
there is non zero row and column then how it rank is 0.. and if there is all number take same number then get rank 1.. why the answer is 0. i dont understand.
Nikhil Saxena said: (11:21am on Thursday 27th April 2017)
i think rank should be 3....because there is no zero in 3 : 3 matrix.
Sufi said: (6:06pm on Sunday 28th May 2017)
Only zero matrix has rank zero

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