linear algebra - For which $n$ the matrix multiplication in $M_n(F)$ is commutative ?

So for $n=0$ and $n=1$, I know matrix multiplication is commutative, but I'm confused as to how to prove that matrix multiplication is not commutative in general for $n\geq 2$. Should I use induction on $n$ or proof by contradiction?

Answer

If you can find a pair of matrices that don't commute for $N$, then for all $n \geq N$ you can take those two matrices as upper left blocks in a matrix where the rest of the columns are fixed ( geometrically, if you've found two linear transformations for dimension $N$, then for any $n > N$, perform those linear transformations on a subspace of dimension $N$ and fix the rest of the space ).

Now just find two matrices of dimension $2$ that don't commute. May I suggest a rotation and a reflection? Is $F$ a general field?