Prove both sufficiency and necessity

[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)

 

[daveb]

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

 

[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.

 

[radou]

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.

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?

 

[mathwonk]

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