Solution
= Solution
The <center of an associative algebra> is the <subalgebra>
$$
Z(A)=\{z\in A:za=az\text{ for every }a\in A\}.
$$
For a pair $(z_1,z_2)$, commuting with every $(a_1,a_2)$ means $z_1a_1=a_1z_1$ and $z_2a_2=a_2z_2$. Varying the two coordinates independently proves both inclusions, hence
$$
\boxed{Z(A_1\times A_2)=Z(A_1)\times Z(A_2).}
$$