The Matrix is the greatest repository of knowledge in the universe. It is, among other things, where all the Doctor Who episodes are stored. Even the missing ones. Even the ones that haven't happened yet. It is the font of the Time Lords' power. It is simultaneously race memory and data storage. It would be completely awesome if they could just manage to keep the Master from getting into it.
The Matrix is, functionally, the souls of dead Time Lords. When a Time Lord dies (or is near death) their accumulated wisdom and experience is added to the Matrix. So are the black-box recordings from TARDISes, which amounts to pretty much the same thing, given that TARDISes are telepathic self-aware 11-dimensional consciousnesses. The Lord President of the High Council of Time Lords can commune with the Matrix via the Circlet of Rassilon (not to be confused with the Coronet of Rassilon, which just gives you mind control powers), and a new President must be accepted by the Matrix in order to assume power. The Matrix is not individuated into consciousnesses, but does seem capable of sending out warnings about things like presidential assassinations. Unless of course it has been hacked. Which is obviously impossible. *shifty glance.*
Of all the ridiculous things Time Lords have invented, the Matrix the one thing that they really got right. Although I might be biased by the fact that it is full of Doctor Who episodes. What a good idea. It also features terrifying clowns, which is rather less of a draw. So maybe I spoke too soon saying the Time Lords got it right. It is in principle impossible for a living mind to enter the Matrix to tamper with it, so naturally this is exactly what happens. It turns out to look like the BBC Quarry, but I'm going to attribute that to a lack of imagination on the part of your average Time Lord. (Omega's anti-matter universe also looks like the BBC quarry, as does Gallifrey itself.) But the quarry is merely the the canvas upon which to paint whatever sur-reality you like.
For this reason the Matrix is the star of the trippiest mind-screwy episodes in the series and it's fantastic. It's Doctor Who's main foray into virtual reality, and so it allows for really different story types. A battle of wits manifested as a physical battle, but where the laws of physics or even narrative convention need not apply. A landscape built of maniacal laughter and shifting reality. Probably the most terrifying thing in "the Deadly Assassin" is when the Doctor clears away the gravel and finds a mirror, reflecting a laughing clown. It's a 5 second scene or less, but it must've inspired a whole generation of new coulrophobes. In "the Ultimate Foe" he fights an even deadlier opponent - bureaucracy. The junior Mr. Popplewick is not allowed to expect anybody... gyeah! Everything is an illusion, but an illusion that can kill. Rather than being reassuring, it means you can't trust anything, least of all your own senses.
Once people are physically, as well as mentally, entering the Matrix (and by people I mean the Doctor) you get the added problem of being potentially unable to tell if you've left it again. I like anything that throws a monkey wrench into the idea of what is real. There is a fan theory that the Doctor never left the Matrix after his second incursion, and that everything that has happened since has been the Valeyard torturing him (which actually accounts remarkably well for the Eighth Doctor's run...) Even from the outside, the Matrix can mess with perceptions of reality by serving as an Unreliable Narrator. When the Doctor is brought in on trial, the evidence is provided by the Matrix, because the Matrix cannot in principle be tampered with. ...except of course that it can be. (My issues with Gallifreyan Law, let me show you them...) But you get this wonderful dynamic of the Doctor arguing with the recording but simultaneously beginning to doubt himself. And we have to doubt everything we've ever watched in the entire show, because the Matrix provides a source for the material we're viewing, and if the Matrix can be hacked, so can Doctor Who continuity itself.
My favorite thing, though, about the Matrix as a Doctor Who viewing source is that it contains episodes that haven't happened yet. Which the Doctor manages to get admitted into evidence in support of the argument "I get better (so please don't execute me)." I... On the one hand, this is excellent science fiction and ridiculously cool. On the other hand this is terrible legal practice. I can't even with the Gallifreyan legal system. Also, I think it really says something about Time Lords that they've got this amazing resource in the Matrix and only the Master has ever thought to explore it. (After the first time they installed a Keeper of the Matrix and a Key and whatnot. It didn't help.)
Well, technically not only the Master. Humanity manages to at last develop Time Travel in the 51st century when the "sleepers" from the Andromeda colony steal its secret from the Matrix. We don't know how they did it, because we first learn about it when the Doctor stumbles upon Ravalox, which is what is left of the Earth after the Time Lords destroyed it in retaliation for the theft. Sure is a good thing they're not around anymore...
But the curious thing is that the Matrix is. Sort of. Maybe. The Nethersphere is described as "a data-slice of the Matrix - Time Lord hard drive." It's not THE Matrix, but it's another instantiation of the same idea, the same technology. A virtual reality and constructed domain that houses the consciousnesses of the dead, but this time still individuated into the people they once were. And, like the Matrix proper, it can interface with living minds - both via some kind of external hookup and directly. There are characters we've seen both inside and outside the Nethersphere. Now who on Earth (...quite...) understands the Matrix well enough to make their own version? Heck, who on Gallifrey does? And like the Matrix proper, the internal and external realities of the Nethersphere can and do affect each other. The question remains can the minds inside the Nethersphere affect its geography, as they can in the Matrix proper? Would it occur to them to try, since they believe it is simply...Heaven? And with different bureaucracy might it actually be? Whatever the answers to these questions, it is once more a much more creative and exciting application of the Matrix than typically occurs to your average Time Lord. It is the Matrix being used at last to its full potential (in one direction at least - it can have multiple full potentials) - the BBC Quarry at last paved over with a stunning cityscape. And its internal geography actually resembles its external geometry - a physical sphere, just one that's a bit bigger on the inside. As the Matrix has always been.
The Matrix is perhaps the coolest thing associated with Time Lords, except possibly the TARDIS herself. It has not been overused in the least, and continues to provide the opportunity for new story types. It messes with reality and is a realm of endless possibility (and rocks). It is Time Lord technology at its very finest and not a ridiculous blunder. It even contains plants. I am so very very happy to have it back at last.
