For all $A,\,B\in M_{mn}$ and for all $1\leq i\leq m$, $1\leq j\leq n$,
\begin{align*}
\matrixentry{A+B}{ij}
&=\matrixentry{A}{ij}+\matrixentry{B}{ij}
&&\knowl{./knowls/definition.MA.knowl}{\text{Definition MA}}\\
&=\matrixentry{B}{ij}+\matrixentry{A}{ij}
&&\text{Commutativity in $\complex{\null}$}\\
&=\matrixentry{B+A}{ij}
&&\knowl{./knowls/definition.MA.knowl}{\text{Definition MA}}
\end{align*}
With equality of each entry of the matrices $A+B$ and $B+A$ being equal Definition ME tells us the two matrices are equal.