How can I apply the concept of singularity to the matrix AB and BA?

[seyma]

Homework Statement

I have a trouble in this proof;
Let A be an m$$\times$$n matrix and B n$$\times$$m matrix. If m$$\neq$$n show that at least one of the matrices AB and BA is singular.

Homework Equations

If it is singular not invertible and det=0 but how can I apply this question?

The Attempt at a Solution

 

[HallsofIvy]

First assume that m< n. Then A is a linear transformation from R^m to Rⁿ with m< n and so cannot map R^m onto Rⁿ: it maps R^m into an at most m dimensional subspace of Rⁿ. B maps all of Rⁿ into R^m. Let v be vector in Rⁿ that is NOT in the image of A. (AB)v= A(Bv) is in the image of A and so cannot be equal to v.

If n< m, reverse A and B.