Matrix Multiplication over Order n Square Matrices is Closed

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

Let $$\mathcal {M}_{R} \left({n}\right)$$ be a $n \times n$ matrix space over $$R$$.

Then matrix multiplication over $$\mathcal {M}_{R} \left({n}\right)$$ is closed.

Proof
From the definition of matrix multiplication, the product of two matrices is another matrix.

The dimensions of an $$m \times n$$ multiplied by an $$n \times p$$ matrix is an $$m \times p$$ matrix, all of whose elements are elements of the ring over which the matrix is formed.

Thus an $$n \times n$$ multiplied by another $$n \times n$$ matrix gives another $$n \times n$$ matrix.

Hence the result.