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Full story — spoilers ahead.
Thomas Anderson is a software programmer who cannot sleep. By day he writes code for a respectable company. By night he is Neo, a hacker chasing a question he cannot phrase: what is the Matrix? He finds strange messages on his computer. He meets Trinity, who already knows his name. Agents in suits want him badly enough to rewrite the rules of a normal arrest.
Morpheus offers him a choice that is really an ending. One pill leaves him in the life he has. The other wakes him up. He wakes up naked in a pod of pink fluid, bald, weak, and plugged into a machine. The year is not 1999. Humanity lost a war with the intelligence it built. The world he remembers is a shared simulation designed to keep bodies calm while they are farmed for power. Morpheus and his crew live on a hovercraft named the Nebuchadnezzar and fight a guerrilla war for a last human city called Zion.
Training is brutal and oddly playful. Neo learns kung fu in minutes because the body is still in a chair and the mind can be overwritten. He meets the Oracle, who tells him he is not the One, then tells him enough that he starts acting as if the opposite might be true. Cypher, a crew member tired of the truth, makes a deal with Agent Smith: plug me back in, wipe the memory, give me a steak that does not exist. The betrayal costs Mouse, Switch, Apoc, and, for a while, Morpheus.
The rescue is the movie’s argument in action. If the world is code, then the rules are habits. Neo and Trinity storm a government building, not because they are commandos, but because they have started to treat gravity as a suggestion. They free Morpheus. Then a subway fight with Smith shows Neo something worse: the agents are programs, and Smith in particular has started to hate the zoo he polices.
At the end Neo is shot in a hallway. Trinity tells a dying man she loves him, and that love is the variable the machines did not budget for. He stands up. Bullets stop. He sees the Matrix as green rain and puts a hand through Smith. He hangs up a phone and promises the machines that he is going to show their prisoners a world without them.
The story is an awakening dressed as a martial-arts picture. Anderson is nobody until he accepts that nobody was the costume. The villain is not only Smith. It is the comfort of a lie that fits. The last shot is not a coronation. It is a man learning he can fly because he finally believes the sky is not real.

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Sydney, Australia
Street-level scenes in the unnamed Matrix city.

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Underground station corridors used for subway sequences.

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Sydney, Australia
Street-level scenes in the unnamed city.

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Downtown grid and phone-booth streets.
Official trailer