2005 Dodge Magnum starts then dies after 3 seconds

Sounds like your Vehicle's Theft Alarm may still be active. Also, like you mentioned, if your key (Transponder) and Sentry Key Remote Entry Module (SKREEM) or Wireless Control Module (WCM) may be defective. The Sentry Key Remote Entry System (SKREES), sometimes referred to as the Sentry Key Immobilizer System (SKIS), authenticates an electronically coded Transponder Key placed into the ignition and sends a valid/invalid key message to the Powertrain Control Module (PCM) based upon the results. The “VALID/INVALID KEY” message communication is performed using a rolling code algorithm via the Programmable Controller Area Network (CAN) data bus. A “VALID KEY” message must be sent to the Powertrain Control Module (PCM) within two seconds of ignition ON to free the engine from immobilization. You should take your vehicle to your local Dealership as they have the proper diagnostic equipment to quickly diagnosis your concern.

that sounds like your security systemis malfunctioning. i had a remote start module installed in my dodge magnum. they messed up programimg the bypass module so it doesn't die when the car is started without the transponder chip. It would died like that .