It could be something as simple as a dying battery. I'm guessing that you drive a Mazda3 so your battery isin't old.
My next question would be what are you running for audio? Is it stock? If not you could be overloading your alternator and not charging the battery enough when you drive. I'd hook up a volt meter to the car and drive around a bit to see what's going on.
I'd also get rid of the inverter and get a proper 12V power supply. 12Vdc -> 120Vac -> 12Vdc never made sense to me.