My hope is that understanding the reason why things are named the way they are in mathematics will help aid in developing mathematical maturity and intuition. Often things are named, and then explain with only the "how" of, not the what or "why". This seems to be the case with cofactors in matrix algebra. Now I now what a factor is, but why is it called "co-factor". What is the "co" part? And as a mathematical object, does it have any meaning on it's own other than being part of a recipe for a solution? Is there a geometric intuition of a cofactor that can explain the What and hence the Why of the name?
The prefix co- come from the Latin where it means ''with'', '' together''. So, in the case of cofactors of a matrix it is justified by the fact that, computing the inverse of a matrix $A$, we have a formula where the entries of the matrix are factors of products, and these other factors co-operates in determining the final result. The prefix co- is used in many different situations in Mathematics ( as co-sine, co-dimension, co-bord, etc.). You can see a brief explanation of this use here.