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Invertibility of the product of matrices

  1. Sep 22, 2016 #1
    1. The problem statement, all variables and given/known data
    Let A and B be n by n matrices such that A is invertible and B is not invertible.
    Then, AB is not invertible.

    2. Relevant equations


    3. The attempt at a solution

    We know that A is invertible, so there exists a matrix C such that CA = I. Then we can right -multiply by B so that CAB = IB = I. Then by the associative property C(AB) = I. By the same argument, we can show that there is a C such that (AB)C = I. So AB has an inverse.

    Obviously this is wrong, because in order for AB to have an inverse, both A and B must have an inverse. So what am I doing wrong?
     
  2. jcsd
  3. Sep 22, 2016 #2
    CAB=IB but you cant conclude IB=I, IB=B
     
  4. Sep 22, 2016 #3

    fresh_42

    Staff: Mentor

    Maybe you should concentrate on the non invertible part. What does it mean to B, not being invertible? Is there a positive property, i.e. without the use of non, not or no?
     
  5. Sep 22, 2016 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    What tools/results are you allowed to use? Do you know about determinants? Do you know how determinants relate to the invertability/non-invertability of a matrix?
     
  6. Sep 25, 2016 #5

    micromass

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    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Assume that ##AB## is invertible. This means that there is a ##C## such that ##CAB = I## and ##ABC = I##. Can you prove now that ##B## is invertible? (and thus deriving a contradiction).
     
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