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Proving Two Matrices to be Equal

  1. Feb 17, 2015 #1

    B18

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    1. The problem statement, all variables and given/known data
    Suppose that A is an m x n matrix and there exists n x m matrices C and D such that CA=In and AD=Im. Prove that
    C=D

    2. Relevant equations


    3. The attempt at a solution
    Im not sure if I'm on the right path here. However my initial thought is that since the matrices are not square there isn't anything to prove by using inverses. So my guess would be i need to use the definition of matrix multiplication on CA=In and AD=Im and try and equate C and D in some way.
     
  2. jcsd
  3. Feb 17, 2015 #2

    jedishrfu

    Staff: Mentor

    But C and D have the same rows and columns so they could be equal. They don't have to be square.
     
  4. Feb 17, 2015 #3

    B18

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    Yes I can see that. But I don't think we can use inverses or inverse properties on this proof because CA and AD are not both the same size matrices.
     
  5. Feb 17, 2015 #4

    Dick

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    You don't have to. Think about the matrix CAD. Use the associative property.
     
  6. Feb 18, 2015 #5

    B18

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    Ok, i think I've got this one nailed down.
    CA=In
    CAD=InD
    C(AD)=D [identity matrix times D is still the matrix D]
    since AD=Im
    we have C(Im)=D
    C=D [identity matrix times C is still the matrix C]
     
  7. Feb 18, 2015 #6

    Dick

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    Nailed.
     
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