It happened just before 2 p.m. at the home on 9667 Laramie Avenue.
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LAPD Commander Stacy Spell said officers responded to the scene after receiving reports that the woman was pointing the gun at her sister and was "actively trying to light their house on fire."
When officers arrived, firefighters were already on the scene, working to put the blaze out.
Footage showed flames and thick smoke coming from the home. A photographer captured on video several LAPD officers arriving to the scene and running toward the house.
Police ended up confronting the woman in the backyard when gunfire erupted. The woman, who was described only as a female in her 50s, was shot by police and later died at the hospital, Spell said.
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It's unclear what prompted the dispute. No other injuries were reported.
Crews were able to extinguish the flames in about 23 minutes, the Los Angeles Fire Department said.
Police are urging anyone who might have information on the incident to come forward.