The woman who was found stabbed to death inside a luxury home has been identified.
Kim Chau, a 39-year-old restaurant worker, was found dead inside a home on Sturt Street in Adelaide's CBD just before 6pm on Monday.
A 21-year-old man was arrested on Tuesday. He had staying at a 'halfway house' for young people.
Police have been scouring rooftops, drains, yards for the possible murder weapon.
Detective Superintendent Des Bray told the ABC Ms Chau appears to have died days before her body was discovered.
Ms Chau was reported missing on Sunday after she didn't show up for work. She was last seen on Thursday night.
Police believe her attacker stole her BMW as it was found behind Australia's Pizza House a short distance from the property.
'What we believe most likely happened is that after the offender has killed the victim, he's returned at some later point and taken the car from Chatham Street.'
Ms Chau was found dead in an upstairs room of the house. She had lived at the property for about a year.
Lois Boswell and Don Frater, who own the property, have expressed their heartbreak over Ms Chau's death.
In a statement, the pair said Ms Chau 'was a friend who has lived with our family over the last year'.
'While we are devastated by these events, there are family members and friends who are primary in our thoughts this morning.'
The luxury, four-bedroom property is in the heart of the city.
Anyone who has any information that may assist the investigation into the woman's death is asked to contact Crime Stoppers on 1800 333 000 or online at www.crimestopperssa.com.au - you can remain anonymous.