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Homework Help: Singularity/Invertibility of Matrix Product

  1. Dec 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that A and B are square matrices of the same order. Prove that if AB is invertible, then A and B are both invertible.

    2. Relevant equations

    3. The attempt at a solution
    I attempted to prove the contrapositive, i.e. if at least one of A,B is singular, then AB is singular. I proved that if B is singular, then AB is singular, but I have not been able to prove that if A is singular, then AB is singular.

    I know that this somehow involves the notion that singular matrices such as C have non-trivial solutions to the homogenous system Cx = 0. But I can't apply it correctly because I don't know the direction to follow.

    Any hints?

  2. jcsd
  3. Dec 3, 2012 #2


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    Science Advisor
    Homework Helper

    If A is singular then Av=0 for some nonzero vector v. Since you've already shown B is nonsingular then v=Bu for some nonzero vector u, yes?
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