\begin{definition}[Definition:Matrix Scalar Product/Ring]
Let $\struct {R, +, \circ}$ be a ring.
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $\struct {R, +, \circ}$.
Let $\lambda \in R$ be any element of $R$.
The '''scalar product of $\lambda$ and $\mathbf A$''' is defined as follows.
Let $\lambda \circ \mathbf A = \mathbf C$.
Then:
:$\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} = \lambda \circ a_{i j}$
Thus $\sqbrk c_{m n}$ is the $m \times n$ matrix composed of the product of $\lambda$ with the corresponding elements of $\mathbf A$.
\end{definition}