Square Matrices over Real Numbers under Multiplication form Monoid

Theorem
Let $\mathcal M_\R \paren n$ be a $n \times n$ matrix space over the set of real numbers $\R$.

Then the set of all $n \times n$ real matrices $\mathcal M_\R \paren n$ under matrix multiplication (conventional) forms a monoid.

Proof

 * Matrix Multiplication is Closed.
 * Matrix Multiplication is Associative.
 * The Unit Matrix is Unity of Ring of Square Matrices.