# Prove both sufficiency and necessity

1. May 29, 2007

### 413

I need some help with this prove

Given that A and B are n x n matrices, prove the following : the product AB is invertible if and only if both A and B are invertible. (Prove both sufficiency and necessity)

2. May 29, 2007

### daveb

What can you say about Det(AB) if it is invertible?

3. May 29, 2007

### mathwonk

can you prove the composition of bijections is a bijection?

and semi conversely, if the composition is a bijection, then the last function is surjective and the first function is injective?

this does most of it. the rest needs some dimension theory, i.e. that a linear map from R^n to itself is injective iff surjective.

the determinant approach is more sophisticated, but very slick and quick.

4. Jun 2, 2007

Just curious, regarding the bijections, are you aiming at the fact that for every matrix there exists a linear operator such that this very matrix is the matrix (of course) representation of that operator?

5. Jun 2, 2007

### mathwonk

yes. multiplication by the matrix is that operator. nread my lin ear algebra book, free online, 15 pages.