Our Matrix is made by Toyota. We have to be sure people know what car they have - you’d be surprised how many people don’t.
You should consider yourself lucky. Pininfarina designed your car. They’ve done the bodywork for some of the most legendary Ferraris in history! 
Your problem sounds a lot like a bad main fuel relay. When their solder joins go bad, they tend to lose contact as they heat up due to expansion. So they work just fine when things are cold, but after it’s warmed up, they start behaving oddly. That’d be where I’d look first, anyway.