Why is matrix multiplication called 'multiplication' if it is non-commutative?

Question

This question begins with the assumption that matrix multiplication was termed 'multiplication' as a form of comparison/parallel to multiplication of integers and real numbers.

Why was matrix multiplication termed 'multiplication' if it does not share the commutativity that other methods of multiplication typically adhere to? And by extension, why was another term not adopted (such as 'matrix application') in respect to this lack of commutativity?

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Edit:

To further clarify, it seems to me that the natural choice for the operation termed 'multiplication' performed with matrices ought to be the Hadamard Product given that it is a direct multiplication of the elements of the matrix (thus retaining the properties of multiplication of real numbers), and that the operation now termed "Matrix Multiplication" should have received a different name because it does not retain the properties exhibited by multiplication in more basic contexts.

So perhaps my question would be better phrased as:

"How did an operation which does not have all the properties of multiplication on real numbers come to be termed 'multiplication'?"

or:

"Why is matrix multiplication called 'multiplication' when the operation it represents seems to have no analogy to multiplication of real numbers?"

Answer 1

Because you multiply the elements of the matrices together. That's the base of the terminology.

Edit:

This answer is old and I don't like it anymore. So here's my new take on the question:

"Multiplication" by itself doesn't need to be commutative (see quaternions). However, one thing that might explain why matrix multiplication is called a "multiplication" is its relation to the identity.

In the world of scalars there's two things you can always say of any number:

1. For any real number $n$, multiplying it by the identity will result in $n$, or $1\times n = n$
2. For any real number $n\neq 0$ there exists a unique number $n'$, such that $n\times n'=1$

When speaking of a square matrix, these both can be easily adapted:

1. Any matrix multiplied by an identity matrix remains the same matrix
2. For any matrix, whose determinant is non-zero, we can find its inverse. Multiplied by this inverse matrix we get the identity matrix

Sadly, these can't be adapted for non-square matrices, since there's no such thing as an $n\times m$ identity matrix. You can still sometimes construct left or right inverses depending on the matrix, but it's not perfect.

Answer 2

An interesting read can be found here: When was Matrix Multiplication invented?

It is interesting to me that determinants have appeared before matrix algebra or even matrices and that the multiplication rule for determinants predates the discovery of matrix multiplication.