Why do Matrices work the way they do? You get taught about matrices and how they work but nobody ever tells you **WHY** they work in the way that they do. What was the idea that sparked the creation of matrices?

Composition of linear transformations. Thus, for example, $$ \begin{align} m & = 4p+7q, & p & =10x-13y \\\ \\\ n & =-3p+2q, & q & = 2x+5y \end{align} $$

So how do you get $m$ and $n$ as functions of $x$ and $y$? You multiply matrices.