# Singular Matrix AB and/or BA

1. Jan 9, 2009

### seyma

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

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

3. The attempt at a solution

2. Jan 9, 2009

### HallsofIvy

First assume that m< n. Then A is a linear transformation from Rm to Rn with m< n and so cannot map Rm onto Rn: it maps Rm into an at most m dimensional subspace of Rn. B maps all of Rn into Rm. 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.