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Properties of Singular Matrices

  1. Jan 30, 2010 #1
    1. The problem statement, all variables and given/known data
    State whether true or false:
    If A and B are singular matrices, then AB is also singular.



    3. The attempt at a solution
    I know that according to the Lemma, if A or B is a singular matrix, then its product AB is also singular. However, it doesn't speak to what happens if two both A and B are singular. I have tried examples using a bunch of singular matrices which I made, and all turned out to be singular. However, I can't get rid of this gut feeling that maybe there is an exception to this situation which I just can't put my finger.
     
  2. jcsd
  3. Jan 31, 2010 #2
    In general, if at least one of them is singular then AB is singular, and that's good enough to prove it.
    Another way to prove this is using determinant.
    Since |A|=0 and |B|=0 (A and B are singular), we get |AB|=|A||B|=0 => AB is singular too.
     
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