Does there exist a formal set-theoretic definition of a matrix?

Question

In mathematics, there are many objects that can be defined purely in terms of sets (e.g. $(a,b)=\{\{a\},\{a,b\}\}$ for ordered pairs, among others). I was wondering if there is a purely set-theoretic definition of a matrices. For example, how could one define the matrix $$\begin{bmatrix}1 & 3 &\pi \\ \cos{(3)} & 33 &0\end{bmatrix}$$ purely in terms of sets? Many thanks.

Answer 1

I remembering having the same question back in the day and I came up with the following definition.

A matrix is just a function of the form $A:I\times J\to R$ where $R$ is a ring. We usually take $I=\{1,\dots,n\}$, $J=\{1,\dots,m\}$ and write $A_{ij}$ instead of $A(i,j)$.