 Solution

The Tarski-Vaught test says that a substructure $A\subseteq B$ is elementary exactly when for every formula $\varphi (x,\bar{y})$ and tuple $\bar{a}\in A$, if $B\models \exists x\varphi (x,\bar{a})$, some $b\in A$ satisfies $B\models \varphi (b,\bar{a})$. Thus every existential assertion with parameters from $A$ that has a witness in $B$ already has a witness in $A$. The matrix may contain quantifiers; restricting it to quantifier-free formulas would not state the full test.