You can visually see structures of your theory (eg, facts about groups) in the patterns of terms in the matrix.
Eg, diagramming a cyclic group ends up with a banding pattern in the portion of your matrix denoting equivalent pairs under the operation… which shows that in some sense, the level maps of + on Zn x Zn form a twisted torus. (Which makes sense in hindsight, as you have the product of two loops with the level maps wrapping around non-trivially.)
That you get ideas such as that by chasing through a purely abstract definition (groups -> diagrams) is wild to me.
Edit:
I actually built a simple case (Z4) in Minecraft — just because I found it surprising.
https://zmichaelgehlke.com/images/z4-plus-map.png
And you get all kinds of weird visuals in diagrams. Like “log” or “exponential” curves trying to diagram a whole field. (Z5, in this case.)