Ring of Square Matrices over Real Numbers

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\struct {\map {\mathcal M_\R} n, +, \times}$ denote the ring of square matrices of order $n$ over $\R$.

Then $\struct {\map {\mathcal M_\R} n, +, \times}$ is a ring with unity, but is not a commutative ring.

Proof
Recall that Real Numbers form Field.

The result follows directly from Ring of Square Matrices over Field is Ring with Unity.