# 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

### radou

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.

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