In one of my other questions (which has no answers by the way - I admit it's rather difficult!), I define a matrix in which each entry is a set. Now that I think about it, I wonder if defining a matrix with set entries is bad practice, because it's not like I can use typical linear algebra tools. I haven't seen many people define matrices like that in past. On the other hand, defining it as such helps me understand the problem a little better. The nice thing about a matrix is it has order in two directions, and it's easy to reference a specific entry, which is helpful in the context of my problem.
Should I avoid doing this practice? Would the Math Illuminati frown upon it?
Here's the link to the question I mention: Equilibrium existence proof
You can certainly define a family of sets indexed by $\{1,\ldots,n\}\times\{1,\ldots,m\}$, and you can write them in a rectangular structure if you have a need to. Nothing at all wrong with this. (And you might well use LaTeX's
matrixenvironments to typeset it).But it probably wouldn't do much mathematical good to call this family a "matrix", unless you have some use for the idea of matrix multiplication, which is the defining feature that makes a matrix a matrix rather than just a two-dimensional array of things.