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Invertibilility of AB given that B is not invertible

  1. Oct 28, 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
    It is easy to show using determinants: det(AB) = det(A)det(B)= 0, so AB is not invertible if either A or B are not invertible.

    Is there an easy way to show this without the use of determinants? I'm just curious
     
  2. jcsd
  3. Oct 28, 2016 #2

    fresh_42

    Staff: Mentor

    If B is not invertible, it has a non-trivial kernel. Take a vector from it and apply AB.
     
  4. Oct 28, 2016 #3
    I see. So then AB has a non-trivial kernel, which means that AB is not invertible.

    What about if we wanted to show that BA is not invertible, given that B is not invertible?
     
  5. Oct 28, 2016 #4

    fresh_42

    Staff: Mentor

    The same. Since A is invertible, we can find a y to an element x of B's kernel, such that x=Ay. Now Bx=0=BAy and y is in the kernel of BA.
     
  6. Oct 28, 2016 #5
    Ah! Makes perfect sense. Thanks
     
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