Why use a matrix?

Why would I want to use a matrix? It's good for organizing a few numbers but I can't find too much use for them. Could someone explain?

We use matrices to study linear transformations between vector spaces. In particular, suppose we have a linear transformation $T: V \to W$ between two vector spaces $V$ and $W$. Suppose we have an ordered basis of $V$ and an ordered basis of $W$. Then we can represent vectors in $V$ and $W$ as column vectors. We can then represent $T$ by a matrix.

Matrices describe how transformations act on a space in a way that we can easily compute with. See this Wikipedia article.