Is Free Will an Illusion?

The idea of determinism, says that everything is predetermined. The idea of free will, which is self explanatory; says that the preceding events are caused by your chosen actions. In the eyes of the incompatibilist if determinism exists, free will cannot. But if free will does not exist, why does it feel as though it does? Is the idea of choice simply that; and idea? Then again there is compatibilism, which says that determinism and free will can coincide. Is that possible? Consider the door example, if you can choose between two doors, Door A and Door B, but unbeknownst to you, Door B is locked. If you choose Door A, is that free will or was it determined that you would go through Door A? In reality you had no other option, but the "illusion" of option was there, whether or not you chose it. So is free will an illusion? I'm really not sure how to answer this.