Can i sue this company?

I bought an alarm for my 2007 tahoe worth 390 dollars, the company then installed the alarm. After the alarm was installed (15 minutes after i received my vehicle) i check everything. The key said that one of my doors were opened which was false, the security light was constantly on, and the back doors when opened while the vehicle was locked didn't sound the alarm. I took it back not even an hour after they installed it and they fixed the light and my key saying my doors where opened. But my security system still doesn't work when the back doors or opened. I told the man that I payed 390 dollars for a working security system, i payed up front they installed it. The system doesn't work, and they are saying i need to take it to another mechanic to fix it. After all that i told them you can take the system out and give me my money back, or make the system work like its suppose too. But they wont do neither, do I have the right to sue? and am I in the right or in the wrong?