Kevin Jewell, 55, shot his wife Jodie dead in their Modbury North property shortly after 6pm on Tuesday, police said.
Superintendent Doc Bray said Ms Jewell, also 55, had previously told a friend she had been the victim of domestic violence, which had not been reported to police.
The friend became concerned when she couldn't contact Ms Jewell and called police who found the mother dead with a gunshot wound in her home.
Kevin Jewell lived on the upper level of the house, while Ms Jewell and her teenage daughter lived downstairs.
"They had been somewhat estranged for a period of time," Supt Bray told reporters.
Their daughter is not believed to have been at home at the time of the incident.
"Obviously she's facing the trauma of what's happened and she's being looked after by relatives," Supt Bray said.
Police on Wednesday obtained an arrest warrant for Jewell in the Adelaide Magistrates Court for the charge of murder.
"If Kevin is watching, I make this appeal to him," Supt Bray said.
"Kevin, you can't undo what has been done last night.
"We are actively looking for you and we will locate you. I encourage you forthwith, straight away, without any delay to engage with a solicitor and negotiate your safe surrender to police."
Supt Bray warned members of the public not to engage with Jewell, who is described as being of average build, 90kg and 178cm tall with a scruffy appearance.
"We believe Kevin is armed with a rifle and ammunition," he said.
Kevin Jewell is believed to be driving a Mitsubishi Triton ute but should not be approached.
"He is considered armed and dangerous and should not be approached by anyone."
Jewell is believed to be driving a white Mitsubishi Triton ute and police believe he could be looking to hide out in the countryside, as he was seen loading items including a backpack and an esky from the home on Tuesday afternoon.
"He likes hunting, he likes camping and he's known to basically set up in the bush in his normal relaxation time, so that's a very strong likelihood," Supt Bray said.
"We've got an enormous amount of resources looking for him.Â
"The alert has gone to every police officer in the state. We've got aircraft looking for him and we've got a large amount of people devoted to finding him."
Supt Bray said Jewell is a licensed firearms holder and his weapons were registered and stored in accordance with the law.
"There's nothing in the antecedents about Kevin or his family that would prevent him from owning a firearm," he said.
Ms Jewell is the fourth woman alleged to have been murdered in South Australia in the past week.
A man was arrested in the remote APY Lands on Tuesday over the murder of a woman known to him at her home in Felixstow in Adelaide's east last week.
It follows the arrests of Michael Edward Amos on Friday and Clifford Aurthor Neumann on Sunday, both charged with murdering women known to them.
None of the matters are believed to be related.
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