We can start observing that there is the »natural«

way to »add« matrices, by adding the components.

So, therefore, this is called »addition«.

I assume that the algebraic laws for matrix

multiplication then relate it to matrix addition

in the way multiplication is related to addition

in a ring (which is the most general algebraic

structure with a multiplication and commutative

addition that I know). So, therefore, it then would

be »the multiplication in the ring of matrices«.

Indeed, I now have confirmed via Wikipedia that

the matrices over a ring are a ring themselves.

s/matrices/NxN matrices/ (i.e. what you say is only true of square

matrices of a particular size).

Overall, the above sounds like a post-justification rather than a

probably explanation of the term. Ring theory started with

Dedekind in about 1870 and it was not until the early 20th century

that rings were unified by axiomatising the abstract structure

they all share.

I get the feeling that Cayley called his composition operation

"multiplication" because he observed that, for square matrices, there

is both a zero (additive and multiplicative) and a multiplicative

identity. It seems unlikely that the term was chosen because of some

deeper algebraic understanding. For example, despite listing 58

properties and theorems about matrices[1] the paper I cited does not

include that fact that multiplication distributes over addition.

[1] Including an unproved version of what came to be called the

Cayley-Hamilton theorem: that a square matrix satisfies its own

characteristic equation p(t) = det(tI - A).