Matrix Multiplication over Order n Square Matrices is Closed
(Redirected from Matrix Multiplication is Closed)
Jump to navigation
Jump to search
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