Answer
Calculate $A+B$ and $B+A$.
Work Step by Step
Consider the matrices with $a_{ij}$ and $b_{ij}$ real numbers:
$$A=\begin{bmatrix}a_{11}&a_{12}&...&a_{1n}\\a_{21}&a_{22}&...&a_{2n}\\...&...&...&...\\a_{m1}&a_{m2}&...&a_{mn}\end{bmatrix}$$
$$B=\begin{bmatrix}b_{11}&b_{12}&...&b_{1n}\\b_{21}&b_{22}&...&b_{2n}\\...&...&...&...\\b_{m1}&b_{m2}&...&b_{mn}\end{bmatrix}$$
First calculate $A+B$:
$$\begin{align*}
A+B&=\begin{bmatrix}a_{11}&a_{12}&...&a_{1n}\\a_{21}&a_{22}&...&a_{2n}\\...&...&...&...\\a_{m1}&a_{m2}&...&a_{mn}\end{bmatrix}+\begin{bmatrix}b_{11}&b_{12}&...&b_{1n}\\b_{21}&b_{22}&...&b_{2n}\\...&...&...&...\\b_{m1}&b_{m2}&...&b_{mn}\end{bmatrix}\\
&=\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&...&a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&...&a_{2n}+b_{2n}\\...&...&...&...\\a_{m1}+b_{m1}&a_{m2}+b_{m2}&...&a_{mn}+b_{mn}\end{bmatrix}.
\end{align*}$$
Then calculate $B+A$:
$$\begin{align*}
B+A&=\begin{bmatrix}b_{11}&b_{12}&...&b_{1n}\\b_{21}&b_{22}&...&b_{2n}\\...&...&...&...\\b_{m1}&b_{m2}&...&b_{mn}\end{bmatrix}+\begin{bmatrix}a_{11}&a_{12}&...&a_{1n}\\a_{21}&a_{22}&...&a_{2n}\\...&...&...&...\\a_{m1}&a_{m2}&...&a_{mn}\end{bmatrix}\\
&=\begin{bmatrix}b_{11}+a_{11}&b_{12}+a_{12}&...&b_{1n}+a_{1n}\\b_{21}+a_{21}&b_{22}+a_{22}&...&b_{2n}+a_{2n}\\...&...&...&...\\b_{m1}+a_{m1}&b_{m2}+a_{m2}&...&b_{mn}+a_{mn}\end{bmatrix}.
\end{align*}$$
Because $a_{ij}$ and $b_{ij}$ are real numbers and in the set of real numbers addition is commutative, we have:
$$a_{ij}+b_{ij}=b_{ij}+a_{ij},\text{ where }i=1,2,...,m,j=1,2,...,n$$
We got that
$$A+B=B+A.$$
