2003 Pontiac Grand Am

The day before christmas I drove to work just fine, and my boss told me I could have the day off, so I went back out to my car and nothing, not even a click of the starter. SO I knew that my battery was weak I could hear it struggling when it would get really cold in the mornings, so I just went accross the street to checker and got a battery, still nothing. So I decide to jump my starter to see if its spinning, and sure enough it was, so im thinking that its an electral problem cause I went back to the seat after that and it started right up. So I drive it home no issues and park in my driveway, shut the car off and start it again, and it started fine. I went back out about 2 hours later and nothing again. SO at no point did I see my securty light flashing for it to be a passlock problem, when I put in my key all my normal dashlights function including the security light but if I attempt to start the car everything goes dark, and then everything turns back on, but not the security light. My buddys camaro was having a simalar problem and I replaced the alternator and its ran great sence. So I went out and got an alternator, put it in and still nothing. Is it my starter or is it the electral system teling the starter not to start?