Definition:Matrix Scalar Product

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 $$\mathbf{A}$$.