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Proving that a matrix is an inverse of another.

  1. Jul 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Let A be an m x n matrix, and suppose there exist 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

    I think it's obvious that C=D=A^(-1). But I'm having trouble proving it since I cannot prove that A is a square matrix and I'm not sure how where to start trying to prove it if A is not square.
     
  2. jcsd
  3. Jul 30, 2012 #2
    Well, WLOG assume that m <= n.

    If AD = Im, then what does that say about A? That means for any m-dimensional vector b, there is an n-dim vector x such that Ax = b (since one can choose x = Db). What does that say about the number of pivot rows of A?

    If CA = In, that means for any n-dimensional vector b, there is an m-dim vector x such that A^Tx = b (one can choose x = C^Tb). What does that say about the number of pivot rows of A^T? What does that say about the number of pivot columns of A?

    Then put those together. :)
     
  4. Jul 30, 2012 #3
    Well, WLOG assume that m <= n.

    If AD = Im, then what does that say about A? That means for any m-dimensional vector b, there is an n-dim vector x such that Ax = b (since one can choose x = Db). What does that say about the number of pivot rows of A?

    If CA = In, that means for any n-dimensional vector b, there is an m-dim vector x such that A^Tx = b (one can choose x = C^Tb). What does that say about the number of pivot rows of A^T? What does that say about the number of pivot columns of A?

    Then put those together. :)
     
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