• Support PF! Buy your school textbooks, materials and every day products Here!

Rank A + rank B <= n if AB=0?

  • #1

Homework Statement


a)Let A and B be nxn matrices such that AB=0. Prove that rank A + rank B <=n.
b)Prove that if A is a singular nxn matrix, then for every k satifying rank A<=k<=n there exists an nxn matrix B such that AB=0 and rank A + rank B = k.


Homework Equations



rank A + dim Nul A = n
Not sure if it's even helpful here.

The Attempt at a Solution


So I am pretty much stuck right now, if someone could point in the right direction, it would be greatly appreciative.

For part (a) I realized that NOT both A and B are invertible, if one of them is invertible, then the other must be the zero matrix so the condition holds. So I was thinking of checking the condition when A and B are not invertible, which doesn't really give me much information to work with.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
rank(B)=dim(range(B)), right? If AB=0 then range(B) has to be contained in null(A), also right? rank A + dim Nul A = n is definitely useful.
 
  • #3
Ah thank you, it's so simple when you put it like that. :D
 

Related Threads on Rank A + rank B <= n if AB=0?

  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
905
  • Last Post
Replies
2
Views
2K
Replies
8
Views
11K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
3
Views
872
Replies
7
Views
507
  • Last Post
Replies
19
Views
9K
Top