Looking for a proof using matrices

[Bachelier]

A,B are nxn. If AB is invertible. Show that A and B are invertible.

I know how to prove it by determinant, using linear transformations and contradictions.
I am looking for a direct way using a proof by matrices. Can anyone think of one?

Thank you.

 

[Outlined]

AB is invertible so there exist a matrix C such that

I = (AB)C = A(BC)

so A is invertible and A^-1 = BC.

Try B yourself.

 

[Bachelier]

I thought about that but I thought it looked too simple.
Thanks. I think this completes all possible ways to prove this problem.
Take a look at my Challenge problem. :)

https://www.physicsforums.com/showthread.php?t=444941"

 

[arkajad]

There is another way if you know that a matrix X is not invertible is equivalent to saying that there is a nonzero vector v such that Xv=0.

Then suppose B is not invertible with Bv=0. Then AB is not invertible since ABv=0. Now, suppose B is invertible but A is not invertible with Av=0. Then ABB^-1 v = 0.

It follows that if A or B are not invertible then AB is not invertible.

 

[Bachelier]

There is another way if you know that a matrix X is not invertible is equivalent to saying that there is a nonzero vector v such that Xv=0.

Then suppose B is not invertible with Bv=0. Then AB is not invertible since ABv=0. Now, suppose B is invertible but A is not invertible with Av=0. Then ABB^-1 v = 0.

It follows that if A or B are not invertible then AB is not invertible.

Good one.