Why is it important to be able to multiply matrices?

Question

I'm not asking how matrix multiplication is done. I know how to multiply matrices and can look at two matrices and say whether or not they can be multiplied. I just want to know what matrix multiplication reveals to us.

Answer 1

This is a pretty broad question, matrix multiplication does many things and I'm sure I cannot give a complete answer as to all the reasons why its useful. Here is one pretty elementary reason why its useful. Recall if I have two functions say $f,g: \mathbb{R}^n \to \mathbb{R}^n$, I can compose them, making a new function $h = f \circ g$. Now if we suppose both $f,g$ are linear operators, and using bases, find a matrix form for them, then $f \circ g = f \cdot g$ as matrices. I encourage you to try to prove this to yourself so you can see its use