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Homework Help: Linear Algebra Inverse mxn

  1. Oct 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove in general that if m does not equal n, then AB and BA cannot both be identity matrices, where A is mxn and B is nxm.

    2. Relevant equations
    None (that I know of at least).

    3. The attempt at a solution
    At first I thought it would be a good idea to define each element in A and B and write out some elements from AB and BA, and hope that I noticed a pattern where I would see something possible only if n=m. This proved very cumbersome and I could not get it to go anywhere.

    Next I tried assuming that both AB and BA equaled identity matrices of appropriate dimensions, with the intention of deriving a contradiction, but I was unfortunately unavle to do so.

    I appreciate any help, as I really don't know what to try next.
  2. jcsd
  3. Oct 22, 2012 #2


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    Science Advisor
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    Gold Member

    Think about what the equations AB = I and BA = I imply, in terms of injectivity and surjectivity of the linear maps represented by A and B.
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