Mate, I had the same problem! Mine's a '97 Verada, and a few days after I bought it from my local Mitsubishi dealer, it just wouldn't turn over when I tried to start it. After RACV failed to get it going, I had it towed to the dealer and demanded they fix it on the spot, which they did. According to the dealership, I had two legit copies of the original key (which the past owner lost and had two spares made).
The dealership said because they were spare keys, they had not been 'coded' properly, which the engine's immobiliser noticed, and after so many times, the engine wouldn't turn over as a security device to make you go to the Mitsubishi dealership from where you brought the car originally, to get your spare keys 'coded' as the original key to stop this from happening again (this all sounded like garbage to me).
So they did this, and off I went. But one week later, after having to pick my brother up from the train station at 1.30am in the morning, the same problem happened, and as you could imagine, I was furious!
I had to catch a cab home, and get up again first thing in the morning to catch a cab back to the station, so I wouldn't get a parking fine, and then call RACV to tow my car back to the Mitsubishi dealership. Now when they saw me again with my car, they almost had a heart attack!
As you could imagine, I was very irate to say the least, and made sure the problem was fixed once and for all, and I stood there for an hour whilst three mechanics practically went over the whole car in front of my eyes, so I knew they weren't going to put the wool over my eyes.
Nearly an hour had passed, and one mechanic pulled out the circuit board for the ignition block, and found that all of the solder had completely worn away after years of use, so no solder was touching the ignition block, so they had to re-solder it.
Finally the problem was solved, and I haven't have a problem since (touch wood). Fortunately I didn't have to pay for the repairs, but at the same time, this should have been fixed before the car was put up for sale (you have to really wonder if car dealerships really actually fix up cars properly before they sell them).
So I hope this helps. Probably the best place to go would be an auto electrics place to get it fixed, if you're still having a problem with it.