Authorities were called to the 900 block of West 41st Street and transported the woman to a local hospital at 2:10 p.m. She died soon after she arrived at the hospital.
Police say the suspect, her grandson, barricaded himself near the crime scene in South Los Angeles at a residence on West 41st St. and Menlo Ave., near Manual Arts Senior High School.
According to family members, the grandson, 42, lives at the residence where the stabbing took place, with his mother and grandmother.
The man's sister, Anika Simpson Richardson, at the scene said her brother has a history of drug problems and may have been hallucinating when he picked up a broom and stabbed grandma in chest.
According to Richardson, the deceased victim was identified as Ethel Porter. Porter's daughter reportedly called 911. Porter was transported to a hospital, where she died.
"As I got over here, my mom told me my brother was hallucinating," said Richardson. "He had said 'My head had been shot off.' He had said that somebody else was killed, and he just picked up a broom and just started stabbing her, stabbing her in her chest. She was already elderly, she lost a lot of blood."
"Right now, the first step is to try to get him to give up peacefully and talk to him verbally," said LAPD Capt. Vito Palazzolo. "We have crisis negotiators en route that will be speaking with him once they make contact with him. That's our first step."
Tactical officers surrounded a shed on the property where the suspect barricaded himself.
By 6 p.m., the suspect had reportedly been taken into custody.
Police said they had not recovered the murder weapon.