Is it possible for both AB and BA to be identity matrices if m does not equal n?

[bologna121121]

Homework Statement

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.

Homework Equations

None (that I know of at least).

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.

 

[jbunniii]

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.