All day on Saturday, Yvonne Woodard and Doyle W Hebert tried to get their 11-year-old grandson to clean up his room at their home in Litchfield Park, Arizona, about 20 miles north-west of Phoenix.

Late that afternoon, the couple, who had custody of the child, sat down in their living room to watch television, the authorities said.

Then, according to a statement from Sergeant Joaquin Enriquez, a spokesperson for the Maricopa County Sheriff’s Office, the boy, whose name was not released, picked up Mr Hebert’s handgun, came up behind the couple and fatally shot his grandmother in the back of the head.

Mr Hebert said he ran after his grandson, but then returned to his wife to try to give her first aid. “Seconds later, Doyle heard another gunshot and saw the grandson take a few steps and then collapse to the ground; the grandson had shot himself,” the statement said. The boy later died.

Mr Hebert retrieved the gun and called 911.

In describing what led to the shooting, the statement merely said:

“They asked their grandson to clean his room and pick up after himself throughout the day, as he was being stubborn about it.”

The statement said the authorities were still investigating, but added there had been no previous signs that the grandson might harm himself or someone else “and there was no cause for concern prior to this event.”

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Mr Enriquez did not reply to questions on Tuesday about what type of handgun had been used, how the child had access to it and whether there would be any charges related to the incident.

Mr Hebert declined to comment when reached by telephone on Tuesday.

School administrators at the Litchfield Elementary School District said in a statement on Monday that psychologists would be at Western Sky Middle School, where the child was enrolled, in Goodyear, a suburb of Phoenix, to help students.

“We are heartbroken to learn of the tragic death of one of our students and his grandmother,” the statement said. “They were valued members of our Wildcat family at Western Sky Middle School and will be deeply missed. Our hearts are with the Hebert family and their loved ones as they endure this unimaginable pain.”

“Nice family,” a neighbour, Walter Venerable, said, according to 3TV, a local news affiliate. “I knew the kid. I mean, he would ride his bicycle around, play around the neighbourhood.”