Abelian Matrices I'm working on some beginner group theory, and I understand that there is an
issue where most matrices are not commutative (i.e. Abelian).  However, I am
interested in solving for the properties of matrices that make them Abelian. 
I was wondering, what is a good way to start exploring the properties of
matrices that could make them Abelian?  I haven't done much of this 
"exploratory math" proofs, but I hear it is good practice.
 A: Provided that you mean by "$A$ an abelian matrix" the following :
$$A\text{ is a matrix  such that } BA=AB\text{ for all matrices }B$$
Usually we say that "$A$ is central".
Then it is indeed an interesting exercise. A first approach would be to write down  the equations (in the coefficients of $A$) given by $AB=BA$ for some $B$'s nd try to understand what is going on. This takes a lot of time and is not the best approach.
The best approach is to think of matrices as endomorphisms of some vector space. Indeed say you work over a field $K$ and matrices of size $n$. 
Define $E:=K^n$, it has a canonical base $e_1,...,e_n$.
If $M$ is a matrix in $M_{n,n}(K)$ then $M$ can be seen as an endomorphism of $E$ by sending $e_i$ to the vector $M.e_i$. 
If you have $AB=BA$ then the associated endomorphisms commute as well. Try now to prove the following lemma :

If $E$ is a $K$-vectorial space, $f$ and $g$ are two linear applications from $E$ to $E$. The relation $fg=gf$ implies that any eigenspace of $f$ is stable by $g$. 

After you have done this you conclude that $A$ (as an endomorphism) stabilizes every eigenspace of every matrix. Uses this to show that $Vect(e_1)$,...,$Vect(e_n)$ and $Vect(e_1+...+e_n)$ are all stable by $A$ and conclude.
