Barely related, but I feel like matrix multiplication isn't "multiplication". It's more like composition.
Eh, what is "multiplication" then? It's a binary operation and it distributes over addition, might as well call it that.
Do you call the composition of two functions a multiplication?
Well, I'm a dynamicist so kinda. By which I just mean that for me the notation f^2(x) generally means f(f(x)) and not [f(x)*f(x)] (exceptions being things like trig functions where the notation is standard but annoyingly inconsistent). But this is maybe confusing the issue...
To get at the heart of it, there are two things at play here:
(i) Terms in math mean whatever they are defined to mean and nothing else
(ii) Terms should ideally be chosen to be useful and consistent
Importantly, "multiplication" does not have a precise mathematical definition as a stand-alone term, so we can't appeal to (i). But there are certain binary operations on certain sets with certain properties that perhaps warrant using the same name for all of them, to highlight exactly those common properties.
On sets that already have a commutative binary operation which we've agreed to call "addition" and have agreed to denote with the symbol "+", we often have another binary operation (which we maybe denote by *) satisfying a*(b+c) = a*b+a*c and (a+b)*c = a*c+b*c. Seems useful and consistent to refer to such operations as "multiplication."
For general functions, composition definitely does not satisfy this property (sin(a+b) =/= sin a +sin b), so it seems less useful to call it "multiplication." Could we? Sure, but then what property is shared by all "multiplications?"
You can debate these things of course. I've avoided using the word "ring" but maybe you want to restrict "multiplication" to only refer to a ring operation. But multiplication in rings must be associative. Is that important to you? Maybe, but then you lose the ability to call the cross product of vectors in R^3 "multiplication." Do you care?
Now of course in the realm of matrices, we have such a binary operation which distributes over addition, so we do tend to call it "multiplication". It also happens to relate to linear transformations of vector spaces, and in a sense coincides with the notion of "composition" in that context (where you should now think about what a useful and consistent definition of "composition" would be).
At the end of the day I'd argue that it's better to talk about multiplying matrices as opposed to composing them. You'd not be insane to take the operation we call matrix multiplication and call it something else, as long as it is defined precisely, but you should think about whether that term is useful and consistent. In this case, you'd be clashing with established nomenclature, though, which is hard to overcome even when it's
not useful or consistent (like sin^2(x) v. sin^(-1)(x)).
As a final quiz, how should we define multiplication of ordered pairs of positive integers? For a,b,c,d positive integers, which is more natural (consistent and useful)?
(i) (a,b)*(c,d) = (ac,bd)
(ii) (a,b)*(c,d) = (ac-bd, ad+bc)
