I recently purchased a 2001 Malibu and when I cut the car off and step out the radio remains on, even after opening door, inwhich I thought it was suppose to turn off. Also my cigarette lighters arent working, checked the fuse and it was good, and suggestions where I should start would greatly be appreciated.

Is this an original or after market radio?

Does the buzzer go when you open the door and the keys are in the ignition?Or the lights are on?? If the are not working either, it is probably the door switch is toast.