Invertibilility of AB given that B is not invertible

[Mr Davis 97]

Homework Statement

Let A and B be n by n matrices such that A is invertible and B is not invertible. Then, AB is not invertible.

Homework Equations

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

 

[fresh_42]

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

 

[Mr Davis 97]

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

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?

 

[fresh_42]

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?

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.

 

[Mr Davis 97]

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.

Ah! Makes perfect sense. Thanks