I just read Baez's very nice blog notes about groupoidification, and around the beginning, he states :
"From all this, you should begin to vaguely see that starting from any sort of incidence geometry, we should be able to get a bunch of matrices. Facts about incidence geometry will give facts about linear algebra!
'Groupoidification' is an attempt to reverse-engineer this process. We will discover that lots of famous facts about linear algebra are secretly facts about incidence geometry!"
There are hints of how this could happen in said notes but (as far as I can see) no concrete example. What would be some nice (easy to begin with) examples of such a phenomenon ?
I'm not looking for particularly hard or even new facts about linear algebra that a groupoidification could unravel (although if there are, I would be very glad to see them); but just some result(s) that appear more naturally, more clearly, or on which another point of view is given via this "groupoidification" method.
So the perfect story would be: "I have this theorem of linear algebra, I know how to prove it but if I were to prove it the usual way there would be a whole lot of annoying mess and non-motivated computations; but there is a nice way to prove it by viewing the situation as some degroupoidification of a problem I know how to handle".
Something that would also be interesting (which is not so different) would be : "Oh I have this fact about incidence geometry/spans of groupoids that's quite nice to prove; and if I translate it into linear algebraic terms I get something interesting, that I might have known but the proof of which is not very enlightening".
If there are a lot of details for your answer and you don't want to write them all down, it's not a problem, I'm fine with just ideas, as long as there's a concrete result in the end.