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

  1. Mar 23, 2010 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations

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

    3. 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.
  2. jcsd
  3. Mar 23, 2010 #2


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    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.
  4. Mar 23, 2010 #3
    Ah thank you, it's so simple when you put it like that. :D
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