Matrix Multiplication over Order n Square Matrices is Closed

Theorem

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

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

Then matrix multiplication (conventional) over $\map {\MM_R} n$ is closed.

Proof

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

The order of an $m \times n$ multiplied by an $n \times p$ matrix is $m \times p$.

The entries of that product matrix 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.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices