I am wondering if anyone can give a simple way to answer the following question in a way that is understandable to a first or second year college student (i.e. one who has not taken a course in abstract algebra).
Typically students are taught two the following 2 basic facts about matrices: Fact 1 : For A, a matrix of size mxp and B, a matrix of size rxn, the matrix product AB is defined iff p=r. Fact 2 : scalars can be considered as 1x1 matrices Given these two facts, the question arises as to how we can say that the products cA and Ac are defined (if m ne 1 and p ne 1 resp for the matrix A described above).
One way around this is to simply define scalar multiplication (as it is normally defined), but this still leads this nagging doubt as to how one resolves this apparent inconsistency that exists with the aforementioned 2 facts.
Sorry for this seemingly stupid question, but it just bugs me a bit.