Square Matrices over Real Numbers under Multiplication form Monoid

Theorem
Let $$\mathcal {M}_\R \left({n}\right)$$ be a $n \times n$ matrix space over the set of real numbers $\R$.

Then the set of all $$n \times n$$ real matrices $$\mathcal {M}_\R \left({n}\right)$$ under matrix multiplication (conventional) forms a monoid.

Proof

 * Matrix Multiplication is Closed.
 * Matrix Multiplication is Associative.
 * The identity matrix is the identity.