Motivation for matrix multiplication.?

Question

I was wondering what the motivation for the definition of matrix multiplication is as it seems somewhat arbitrary. I'm not talking about a row matrix times a rectangular matrix, or a column matrix times another rectangulart matrix. I'm talking about the general case. If there is a theoretical reason for the definition please inform me. If the definition is motivated by something practical it would be nice to know of aswell(noth would be best).

Are there any other motivations or theoretical reasons?just out of curiousity.

Answer 1

Sure. When you multiply two matrices you are combining two linear transformations.

ex
You have a rotation matrix,A and a projection matrix, B

The AB is projection then rotation
and
BA is rotation then projection

Matrix multiplication is not communitive in general

Answer 2

Multiplying matrices does seem kind of weird, but makes sense in the context of vector spaces. A matrix defines a vector space so multiplying two matrices would amount to translating from one vector space into another. This is easy enough to see in the case of a row or column matrix, because we're used to thinking in three dimensions. In higher dimensions most people can't visualise it but the principle is the same. This is how mathematicians and theoretical physicians can postulate higher order dimensional space even though most people, including scientists, have no idea what it would "look" like. As long as you can get the mathematical operations to remain internally consistent, you can go far beyond what we as humans normally experience.