Sum of Real Matrices

Let $m$ and $n$ be natural numbers, and let $A$ and $B$ be real $m\times n$ matrices, with entry notation and index sets $[m]$ and $[n]$ as there. Addition of real numbers is the addition of the ordered field $\mathbb{R}$.

The sum $A+B$ is the real $m\times n$ matrix whose entries are

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