Definition:Matrix Scalar Product

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

Let $\mathbf A = \left[{a}\right]_{m n}$ be an $m \times n$ matrix over $\left({R, +, \circ}\right)$.

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:
 * $\forall i \in \left[{1 \,.\,.\, m}\right], j \in \left[{1 \,.\,.\, n}\right]: c_{i j} = \lambda \circ a_{i j}$

Thus $\left[{c}\right]_{m n}$ is the $m \times n$ matrix composed of the scalar product of $\lambda$ and the corresponding elements of $\mathbf A$.