I recently watched the Matrix and I was wondering whether it depends on the philosophy put forth by Plato or by that put forth by Aristotle. I noticed there were several analogies to Plato's analogy of the cave, mainly, the Matrix is the shadows on the cave, the machine controlled world is the cave and Zion is the outside world. With Aristotle, I noticed that the four causes can find examples in the Matrix. Mainly, the final cause of the simulation is to keep humans happy while the act as batteries, the formal cause is the 20th century world, the material cause is thought and the efficient cause is rather left to imagination. But I was wondering, which of these is correct, or to which philosophy does the Matrix bear more resemblance to or which one is more inherent of the plot.
Certainly we think first of Plato's Cave when envisioning a world of illusion, a phenomenological prison. But the famous cave metaphor does not entail an intentional deception with some rational agency behind it, as with Descartes Demon or Althusser's state idealogical apparatus.
On the other hand, the political structure of the Republic itself does requires a guardian class of "philosopher kings" with a highly developed level of mathematical reason, a social myth based on the noble lie, and a relative lack of self-interest apart from their own reproduction. Computers could qualify.
Yet Plato's rulers retain their moral allegiance to the citizens, the welfare of the state, and the Good. In the Matrix the class division has become logically antithetical and utterly predatory, the rulers operate logically to increase their power by the total appropriation of the vital forces reproduced in humans. A "police" boundary of alienation, terror, and ignorance separates the two classes. The ruling class acts as a continuous unity, while the ruled are helplessly encased in their discrete bodies.
Thus, a radical technological extrapolation of humanist Marxism and "culture industry" critique might be the better analogy. And of course, movies like the Matrix play their own role in this ideology. By equating Plato's socialist hypothesis of the just state with demonic totalitarianism, they promote a libertarian "heroic" ideal that fits all too well into the benighted individualism of the commodity system. So indeed, as Plato warns, beware the dramatists tugging on heartstrings.
The Matrix is surely more Platonic than Aristotelian. In general, Plato believed that reality could turn out very different than what it seemed to be at the time. Aristotle believed that reality was, by and large, similar to what it seemed to be at the time. Plato believed that only few human beings could free themselves sufficiently and see the truth. Aristotle believed that this ability was much more commom. The later Plato was even more skeptical than the earlier Plato. Also Plato's academy became, after his death, disposed to radical skepticism.
The fact that Aristotle's four causes are applicable to the Matrix is inconsequential here. The four causes are applicable both in and out of the Matrix, so that they are not evidence for either possibility.
In addition to the other excellent answers, remember the scene in The Matrix (1999) when Neo hide his money inside an hollowed book entitled Simulacra and Simulation.
That book was written by french post-modern philosopher Jean Baudrillard and published in 1981 (who was inspired and deeply influenced by Marx). Here, Baudrillard argues that twentieth-century society has replaced all reality and meaning with symbols and signs, and that human experience is a simulation of reality.
You can interpret The Matrix as a criticism of the consumer culture, a culture saturated with commercial advertising that may be distracting us from the reality. And consequently, the simulacra of the real become more real than reality (or "hyper-real"). In the mean time humans are are being exploited by someone, just as the machines exploit the humans in the Matrix. So, this is related with Marxism where the working class is exploited by the ruling classes because it does not perceive itself as being exploited.
The movie The Matrix is most typically associated with the philosophy of Descartes, specifically, the part of Meditation I where he imagines that all he experiences is an illusion created by a malicious demon.
However, Descartes is an Idealist (philosopher who believes that ideas and mental concepts represent a more fundamental level of reality than material objects) and Idealism in Western philosophy traces back to Plato, who believed that the entire material world as we experience it is just an imperfect copy of a more true, beautiful, eternal and good world composed entirely of ideals. So in this sense, The Matrix does have a strong connection to Plato, and as you mentioned, its central conceit --of a populace lulled into complacency by belief in a set of illusionary images --is strongly reminiscent of Plato's Allegory of the Cave.
Any connections to Aristotle are much more tenuous, particularly since Aristotle was focused on the world as we experience it, rather than on some deeper level of reality beneath it. The "Four Causes" are intended to apply broadly; there is no specific relationship between them and the Matrix's metaphysics.