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Singular Matrix AB and/or BA

  1. Jan 9, 2009 #1
    1. The problem statement, all variables and given/known data
    I have a trouble in this proof;
    Let A be an m[tex]\times[/tex]n matrix and B n[tex]\times[/tex]m matrix. If m[tex]\neq[/tex]n show that at least one of the matrices AB and BA is singular.

    2. Relevant equations

    If it is singular not invertible and det=0 but how can I apply this question?

    3. The attempt at a solution
     
  2. jcsd
  3. Jan 9, 2009 #2

    HallsofIvy

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    First assume that m< n. Then A is a linear transformation from Rm to Rn with m< n and so cannot map Rm onto Rn: it maps Rm into an at most m dimensional subspace of Rn. B maps all of Rn into Rm. Let v be vector in Rn that is NOT in the image of A. (AB)v= A(Bv) is in the image of A and so cannot be equal to v.

    If n< m, reverse A and B.
     
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