The discussion revolves around proving that the product of two n x n matrices, AB, is invertible if and only if both matrices A and B are invertible. Participants explore both sufficiency and necessity in their proofs, as well as related concepts in linear algebra such as the properties of bijections and determinants.
Participants express various viewpoints on the proofs and concepts related to matrix invertibility and bijections, indicating that multiple competing views remain without a clear consensus on the best approach or proof method.
Some discussions reference the need for dimension theory and properties of linear maps, which may depend on specific definitions or assumptions that are not fully articulated in the thread.
Readers interested in linear algebra, particularly those studying matrix theory, properties of determinants, and the relationship between linear transformations and bijections.
mathwonk said: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.