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If A,B are nxn and AB is invertible, then A and B are invertible

  1. Feb 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Let A and B be nxn matrices such that AB is invertible. Show that A and B are also invertible.

    2. Relevant equations


    3. The attempt at a solution
    I feel that there are many ways to do this. My first idea was to use the fact that CAB = I and ABC = I for some C, which implies that (CA)B = I and A(BC) = I, which means that A has a right inverse and B has a left inverse. But since A and B are nxn, BC and CA are not just right and left inverses, respectively, but they are the inverses of A and B. Is this correct reasoning?

    Also, would it be any better to make use of the isomorphism from matrices and linear transformations, prove the result for linear transformations, and hence prove the result for matrices?
     
  2. jcsd
  3. Feb 14, 2017 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Do you know about determinants?
     
  4. Feb 14, 2017 #3
    I know about them, but in this course we are not allowed to use them yet.
     
  5. Feb 15, 2017 #4

    fresh_42

    Staff: Mentor

    You could show, that the kernels of ##A## and ##B## have to be ##\{0\}## and apply the dimension formula ##\dim \ker C + \dim \textrm{im}\, C = n## to show surjectivity.
     
  6. Feb 15, 2017 #4

    pasmith

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    Homework Helper

    This is correct; it is also the easiest way to prove the result.
     
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