Matrix Multiplication is not Commutative/Order 2 Square Matrices

Theorem
Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\map {\mathcal M_R} 2$ denote the $2 \times 2$ matrix space over $R$.

The operation of (conventional) matrix multiplication is not commutative over $\map {\mathcal M_R} 2$.

Proof
As $R$ is a ring with unity, we have that:

Now let:

By definition, both $\mathbf A$ and $\mathbf B$ are elements of $\map {\mathcal M_R} 2$.

It will be demonstrated that $\mathbf A$ and $\mathbf B$ do not commute.

We have:

and:

and it is seen that:
 * $\mathbf A \mathbf B \ne \mathbf B \mathbf A$

Thus, whatever the nature of the ring with unity $R$, it is never the case that matrix multiplication is commutative over $\map {\mathcal M_R} 2$.