Proving Two Matrices to be Equal

[B18]

Homework Statement

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

Homework Equations

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.

 

[jedishrfu]

But C and D have the same rows and columns so they could be equal. They don't have to be square.

 

[B18]

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.

 

[Dick]

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.

You don't have to. Think about the matrix CAD. Use the associative property.

 

[B18]

You don't have to. Think about the matrix CAD. Use the associative property.

Ok, i think I've got this one nailed down.
CA=I_n
CAD=I_nD
C(AD)=D [identity matrix times D is still the matrix D]
since AD=I_m
we have C(I_m)=D
C=D [identity matrix times C is still the matrix C]

 

[Dick]

Ok, i think I've got this one nailed down.
CA=I_n
CAD=I_nD
C(AD)=D [identity matrix times D is still the matrix D]
since AD=I_m
we have C(I_m)=D
C=D [identity matrix times C is still the matrix C]

Nailed.