Linear algebra proof - inverses

[miky87]

Homework Statement

Let m < n. Let A be an n × m matrix, and let B be an m × n matrix. Prove that AB /= In .

Homework Equations

The Attempt at a Solution

since m<n, the reduced form of matrix B will have free variables.
I know that if A and B are invertible matrices, AB will be as well but does the converse also hold?

 

[micromass]

Hi miky87!

I know that if A and B are invertible matrices, AB will be as well but does the converse also hold?

This makes no sense. A matrix can only be invertible when it's square. And these are not square matrices.

What can you tell us about the rank of A, B, AB and I_n??

 

[Ray Vickson]

The question make perfect sense: he is being asked to show that A*B cannot be the identity matrix (i.e., that a non-square matrix cannot have a left- or right-inverse). This seems closely related to another question in this Forum, and the same advice given there applies here.

RGV

 

[micromass]

The question make perfect sense: he is being asked to show that A*B cannot be the identity matrix (i.e., that a non-square matrix cannot have a left- or right-inverse). This seems closely related to another question in this Forum, and the same advice given there applies here.

RGV

A matrix A is invertible if and only if there is a matrix B such that AB=BA=I. This can only be satisfied for square matrices. So calling A and B invertible in his question does not make any sense.
Left and right inverses do make sense, but I doubt he meant that.