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System of equations

Yankel

Active member
Jan 27, 2012
398
Hello guys

I need an advice on this one...

A is a matrix of 4X8, and b is a vector of 4X1. It is known that the system Ax=b has infinite number of solutions.

Which statement is correct ?

a. rank(A)=4
b. rank(A)>4
c. rank(A|b)=rank(A)
d. rank(A|b)=3
e. rank(A|b)>rank(A)

my intuition say c, but I don't know why...
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Hello guys

I need an advice on this one...

A is a matrix of 4X8, and b is a vector of 4X1. It is known that the system Ax=b has infinite number of solutions.

Which statement is correct ?

a. rank(A)=4
b. rank(A)>4
c. rank(A|b)=rank(A)
d. rank(A|b)=3
e. rank(A|b)>rank(A)

my intuition say c, but I don't know why...
Hi Yankel,

Your intuition is correct. There is a theorem saying that a system is consistent (has one or more solutions) if and only if the rank of the augmented matrix (A|b) equal to the rank of A. Refer: http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node25.html

Answer b is incorrect as the rank of a matrix cannot exceed its dimensions. i.e: $rank(A)\leq min\{4,8\}=4$
If the system has infinite many solutions it is consistent. Hence by the above theorem answer e is also incorrect. The remaining two answers (a and d) are not necessities for the consistency.