# Invertibilility of AB given that B is not invertible

1. Oct 28, 2016

### Mr Davis 97

1. The problem statement, all variables and given/known data
Let A and B be n by n matrices such that A is invertible and B is not invertible. Then, AB is not invertible.

2. Relevant equations

3. The attempt at a solution
It is easy to show using determinants: det(AB) = det(A)det(B)= 0, so AB is not invertible if either A or B are not invertible.

Is there an easy way to show this without the use of determinants? I'm just curious

2. Oct 28, 2016

### Staff: Mentor

If B is not invertible, it has a non-trivial kernel. Take a vector from it and apply AB.

3. Oct 28, 2016

### Mr Davis 97

I see. So then AB has a non-trivial kernel, which means that AB is not invertible.

What about if we wanted to show that BA is not invertible, given that B is not invertible?

4. Oct 28, 2016

### Staff: Mentor

The same. Since A is invertible, we can find a y to an element x of B's kernel, such that x=Ay. Now Bx=0=BAy and y is in the kernel of BA.

5. Oct 28, 2016

### Mr Davis 97

Ah! Makes perfect sense. Thanks