Proving C equals D when CA = In and AD = Im

[phosgene]

Homework Statement

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.

Homework Equations

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.

 

[who_]

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. :)

 

[who_]

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. :)