The violence is mostly bloodless with shootouts and stylized martial arts combat that result in little to no blood.
An Agent is shot in the head by Trinity. There is no blood/gore
Lots of killings with machine guns, resulting in numerous bodies and little blood.
There are gunshot wounds on the chests and on the backs of a few people.
One man gets zapped by a laser gun
There are numerous martial arts fights with lots of punching and hitting. A few more intense scenes have closeups of people being punched with blood around their mouths.
Some people get their bodies slammed against walls. Not very brutal.
A man has a creature extracted from his navel, no blood is shown.
Trinity throws a knife into a man's forehead during an action scene, some blood is shown if you pause in the exact moment where he collapses
An agent is shot at an extremely close range; we don't see the impact in detail but there is a brief red mist.
A man is shot by a lot of bullets. You briefly see the the impact.
Neo shoots a few people at a security gate.
When people die in the matrix, their bodies in the real world cough up blood and die as well.
A couple of people are killed with a laser gun, we don't see much detail.
A woman beats another man up, she face-palms him and kicks him. She dodges bullets discharged by other cops, throws a chair at one of them, shoots another, then chops/kicks the final one to the ground. It is assumed all those cops die. She flees from more cops, jumping off rooftops and crashing through windows, and runs to a pay phone in order to get out of the situation. A garbage truck plows down the phone booth but she lives.
There is a closeup of Morpheus being shot in the leg with a tiny bit of blood spray. Not too graphic.
A group of police officers viciously beat Morpheus and he is seen covered in dust and has blood on his face. Morpheus and Agent Smith fight briefly. Morpheus gets punched several times. Blood is seen around his mouth.