The article/webpage is a nice walk-through for the uninitiated. Half the challenge of doing matrix calculus is remembering the dimension of the object you are dealing with (scalar, vector, matrix, higher-dim tensor).
Ultimately, the point of using matrix calculus (or matrices in general) is not just concision of notation but also understanding that matrices are operators acting on members of some spaces, i.e. vectors. It is this higher level abstraction that makes matrices powerful.
For people who are familiar with the concepts but need a concise refresher, the Wikipedia page serves well: