Doing It for the Art: While watching the "Lady in the Red Dress" scene, you might notice the same people are passing Neo and Morpheus more than once. Production goof? Nope! All of the extras in the scene are actual twins. They searched all over Sydney for twins and brought them in to create the impression that Mouse, while writing the Agent training program, got lazy and just copy-pasted the same models over again instead of making unique ones. Did you notice that... or were you staring at the lady in the red dress?
Edited for Syndication: Basic cable cuts of the film frequently substitute "Shucks" or "Smokes" for "Shit", and change Neo's line in the interrogation room scene to "How about I give you a flipper?", while completely cutting the actual middle finger (and, by consequence, the "You give me my phone call" part of the line). Also "Jeepers creepers (originally "Jesus Christ"), that thing's real?!!!"
Executive Meddling: Dropping the original idea of Human Resources being a vast neural net for the more literal concept of human Duracell batteries because Viewers Are Morons. Giving all the other stuff that has to be explained (or not) in this trilogy, one wonders why this idea was so difficult to put across.
Originally, Will Smith was offered the role of Neo, but turned it down for his part in Wild Wild West. To his eternal woe.
The "mental projection" of the Matrix would have been plot-relevant, with the butch blonde character "Switch" appearing as a woman in the Matrix, but a man in the real world. This is perhaps mirrored by Larry AKA Lana Wachowski coming out as a transgendered woman in 2012.
The Wachowskis were given the choice of making the Matrix sequels or Batman Begins. Imagine that...
Jet Li and Michelle Yeoh were initially offered the role of Seraph; Li, who the role was written for, asked for an equal salary to Reeves's and didn't get it, while Yeoh had scheduling conflicts.