Square Matrices over Real Numbers under Multiplication form Monoid

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

Then the set of all $n \times n$ real matrices $\map {\mathcal M_\R} n$ 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.