Undecidable language and Turing Machines
To my understanding this turing machines accept states, are all those that were of the rejection states in the first Turing Machine.


You presume to know something about how the machine for $\overline{A}$ was built; don't, because you can't.

What's more, this construction (flipping accepting states) does not work for TMs that do not always halt, as any one for $A$ has to be.

Thus, a semi-decider for $\overline{A}$ has to look differently from what you describe.



Regarding the original problem you were given, it is a weird formulation to test your understanding of basic closure properties.

Hint: Assume there was a semi-decider for $\overline{A}$. What follows about $A$?