# Square Matrices over Real Numbers under Multiplication form Monoid

## Theorem

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

Then the set of all $n \times n$ real matrices $\map {\MM_\R} n$ under matrix multiplication (conventional) forms a monoid.

## Proof

Matrix Multiplication over Order n Square Matrices is Closed.
Matrix Multiplication is Associative.
The Unit Matrix is Unity of Ring of Square Matrices.

$\blacksquare$