Yes if $A$ and its complement are accepted by Turing machines.then $A$ is recursive.
Suppose a language $A$ is recursively enumerable. That means there exists a Turing machine $T1$ that, given any string of the language, halts and accepts that string.
Now let's also suppose that the complement of $A$, $A'= \{w: w \mid A\}$, is recursively enumerable. That means there is some other Turing machine $T2$ that, given any string of $A'$ halts and accepts that string. So any string belongs to either $A$ or $A'$. Hence, any string will cause either $T1$ or $T2$ (or both) to halt. We construct a new Turing machine that emulates both $T1$ and $T2$, alternating moves between them. When either one stops, we can tell (by whether it accepted or rejected the string) to which language the string belongs.
Thus, we have constructed a Turing machine that, for each input, halts with an answer whether or not the string belongs to $A'$. Therefore $A$ and $A'$ are recursive languages.