Scalar Multiple of a Real Matrix

Let $m$ and $n$ be natural numbers, let $A$ be a real $m\times n$ matrix, with entry notation and index sets $[m]$ and $[n]$ as there, and let $\lambda\in\mathbb{R}$, where $\mathbb{R}$ is the set of real numbers with the multiplication of its ordered field structure.

The scalar multiple $\lambda A$ is the real $m\times n$ matrix whose entries are

for all $i\in[m]$ and $j\in[n]$.