I'm nearing the end of my first year of Calculus and am pretty confident in the parts of it I've learned, yet I still don't have a good understanding of matrices, which seem like they should be easier to understand and work with. They were never formally taught in any of the math courses my school has given me and were only ever briefly mentioned.
I don't fully understand what they represent or the logic behind how they are multiplied. Why are the left matrix's rows multiplied by the second matrix's columns to form columns in the resulting matrix? Why not columns by rows to form rows? Was it an arbitrary decision that was made or does it fit what concept of the matrix?
I could memorize how to perform operations on matrices and how to use them to solve certain problems, but what I'm trying to do is understand them conceptually.
Specifically, I was reading a page about the matrix representation of 2d transformations as used by CSS3 and found it all very confusing.