Can the Sum of Matrix Ranks Be Greater Than n When AB Equals Zero?

[calorimetry]

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.

 

[Dick]

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.

 

[calorimetry]

Ah thank you, it's so simple when you put it like that. :D