[AndyGoode]

Matrices make everything a lot neater,  everything comes down to a sum of products, they are great.

The first time I ever used matrices on my own project was when I wanted to solve a system of simultaneous equations, and I remembered the Gaussian elimination method, even a few years out of college. I forgot exactly what I was working on, but it was something recreational, like analyzing personal tastes, or analyzing dating services, or something like that.

Data structures are pretty amazing in themselves--trees, graphs, linked lists. Heck, even molecules and chess attack lines are basically data structures. The possibilities are endless. Basically you just come up with your own shape [data structure], then figure out how to combine other copies of that shape, especially in a way that resembles addition or multiplication to some extent. Also often very useful is to add metrics or functions to perform on those data structures, similar to determinant, trace, eigenvector, eigenvalue, etc. as done on matrices. If things fall into place to the extent that you can predict or discover things you couldn't before, then you know you're really onto something.

After posting that chart above of which matrices are compatible, I decided to treat horizontal [= transposed] vectors as a third type of  possible object to multiply, which extended my 2x2=4 list to a 3x3=9 list, which I've attached below. That would probably be a good teaching tool. I've never seen any math teacher make a chart/handout like that, maybe because it's so 'obvious'.