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Prove both sufficiency and necessity

  1. May 29, 2007 #1

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    I need some help with this prove

    Given that A and B are n x n matrices, prove the following : the product AB is invertible if and only if both A and B are invertible. (Prove both sufficiency and necessity)
     
  2. jcsd
  3. May 29, 2007 #2
    What can you say about Det(AB) if it is invertible?
     
  4. May 29, 2007 #3

    mathwonk

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    can you prove the composition of bijections is a bijection?

    and semi conversely, if the composition is a bijection, then the last function is surjective and the first function is injective?

    this does most of it. the rest needs some dimension theory, i.e. that a linear map from R^n to itself is injective iff surjective.

    the determinant approach is more sophisticated, but very slick and quick.
     
  5. Jun 2, 2007 #4

    radou

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    Just curious, regarding the bijections, are you aiming at the fact that for every matrix there exists a linear operator such that this very matrix is the matrix (of course) representation of that operator?
     
  6. Jun 2, 2007 #5

    mathwonk

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    yes. multiplication by the matrix is that operator. nread my lin ear algebra book, free online, 15 pages.
     
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