Shutter Island

In 1954, two U.S. Marshals â Edward "Teddy" Daniels and his new partner, Chuck Aule â travel to the Ashecliffe Hospital for the criminally insane on Shutter Island located in Boston Harbor. They are investigating the disappearance of patient Rachel Solando, who was incarcerated for drowning her three children. Their only clue is a cryptic note found hidden in Solando's room: "The law of 4; who is 67?" They arrive just before a storm hits, preventing their return to the mainland for a few days.
Daniels and Aule find the staff confrontational; Dr. John Cawley, the lead psychiatrist, refuses to turn over records, and they learn that Solando's doctor, Dr. Lester Sheehan, left the island on vacation immediately after Solando disappeared. Though they are given access to the hospital, they are told Ward C is off limits, while the lighthouse has already been searched. Whilst being interviewed, one patient secretly writes the word "RUN" in Daniels' notepad. Daniels starts to have migraine headaches from the hospital's atmosphere and experiences waking visions of his involvement in the Dachau liberation reprisals. He has disturbing dreams of his wife, Dolores Chanal, who was killed in a fire set by Andrew Laeddis, a local arsonist. In one instance Chanal tells Daniels that Solando is still on the island somewhere. The same applies to Laeddis who everyone claims was never there to begin with. Daniels later explains to Aule that locating Laeddis was an ulterior personal motive for taking the case.
During their investigation Daniels and Aule find that Solando has abruptly resurfaced with no explanation as to her former whereabouts or how she escaped. This prompts Daniels to break into the restricted Ward C. There he encounters George Noyce who is a patient in solitary confinement. Noyce warns Daniels that the doctors are performing questionable experiments on the patients, some of whom are taken to the lighthouse to be lobotomized. Noyce warns Daniels that everyone else on the island, including Aule, is playing an elaborate game specifically designed for Daniels. Daniels regroups with Aule and is determined to investigate the lighthouse. While climbing the cliffs toward it they become separated, and Daniels later sees what he believes to be Aule's body on the rocks below. By the time he climbs down the body has disappeared, but he finds a cave where he discovers a woman in hiding who claims to be the real Rachel Solando. She states that she is a former psychiatrist at the hospital who discovered the experiments with psychotropic medication and trans-orbital lobotomy in an attempt to develop mind control techniques. Before she could report her findings to the authorities, however, she was forcibly committed to Ashecliffe as a patient. Daniels returns to the hospital, but finds no evidence of Aule ever being there.
Convinced that Aule was taken to the lighthouse, Daniels decides to break into it, only to discover Cawley calmly waiting there for him. Cawley explains that Daniels is actually Andrew Laeddis, "[their] most dangerous patient", incarcerated in Ward C for murdering his manic depressive wife, Dolores Chanal, after she drowned their children, thus revealing that "Solando" is actually his deceased wife. Edward Daniels and Rachel Solando are anagrams of Andrew Laeddis and Dolores Chanal ("the law of 4"), and Laeddis is the 67th patient at Ashecliffe ("who is 67?"); furthermore, the little girl from Laeddis' recurring dreams is his daughter, Rachel.
According to Cawley, the events of the past several days have been designed to break Laeddis' conspiracy-laden insanity by allowing him to play out the role of Daniels. The hospital staff, including Dr. Sheehan posing as Aule and a nurse posing as Rachel Solando, were part of the test, and the migraines that Laeddis suffered were withdrawal symptoms from his medication, as well as hallucinating the "real Rachel Solando". As memories of reality and realization that Daniels was only his play, this overwhelms Laeddis, and he faints.
Laeddis awakes in the hospital, under watch of Cawley and Sheehan. When questioned, Laeddis tells the truth in a coherent manner, which satisfies the doctors as a sign of progression. Nevertheless, Cawley notes that they had achieved this state nine months before but Laeddis had quickly regressed, and further warns that this will be Laeddis' last chance to redeem himself.
Some time later, Laeddis relaxes on the hospital grounds with Dr. Sheehan, but calls him "Chuck" and says they need to leave the island. Sheehan shakes his head to an observing Cawley, who gestures to the orderlies towards Laeddis. As he is about to be taken away (presumably to have a lobotomy), Laeddis asks Dr. Sheehan, "Which would be worse? To live as a monster, or die as a good man?" suggesting his insanity is only a ruse to get the lobotomy, as he can no longer live with himself knowing what happened.

Question: Who did Andrew murder?
Answer: His wife.