Square Matrices over Real Numbers under Multiplication form Monoid

Theorem
Let $$\left({R, +, \circ}\right)$$ be a ring with unity.

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

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

Proof

 * Matrix Multiplication is Closed.


 * Matrix Multiplication is Associative.


 * The identity matrix is the identity.