You're joking, right? You probably know that for matrix reloaded alone, they had more cash than Jackson for his whole trilogy? limited cash my ass.
As for convincing New Line, there was no such thing. They would agree, or they wouldn't. It's not coz Jackson would say "come on, it's gonna be great!" that they will let him do it. They studied the possibilities, and they decided it would work. That's it.
If matrix was planed as a trilogy from the begining, everybody would have known it before the first movie started. It's exactly the same deal than with Starwars. In each case, the first movie works well as a standalone ( and in the matrix case, it still works well as a stand alone movie

). They left it open ended in case it would be a hit, but that's it. Matrix plot was well thought, the 2 and 3 are rushed. I repeat: RUSHED. They would have spent more time on the plot, and less on the CGI, the movies would have been better.