1996 Pontiac Grand Prix

The car would periodically refuse to start, the radio and everything else would work fine, but nothing from the starter. No clicking, nothing when you would turn the key. I would wait a while, then try it and it would start up beautifully. Now it will not start at all. We took the starter, alternator, and battery in and had them tested, they all passed. We put in a new ignition switch and starter relay, still nothing. I wanted to try something, so I ran a wire from the battry directly to the starter solenoid, then the car started, but died after a secondo or two, not like a sputtering, ' I ran out of gas', more like someone hit a kill switch, does anyone have any suggestions?