Matrix Multiplication is Closed

Theorem

Let $\struct {R, +, \circ}$ be a ring.

Let $\map {\mathcal M_R} n$ be a $n \times n$ matrix space over $R$.

Then matrix multiplication (conventional) over $\map {\mathcal M_R} n$ 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 entries are elements of the ring over which the matrix is formed.

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

Hence the result.

$\blacksquare$