The death toll in a Northern California shooting rampage rose to five Wednesday after authorities said they found the body of the gunman’s wife hidden under the floor of the couple’s house. The count, authorities say, could have been much worse had an elementary school not ordered an immediate lockdown.

The gunman, identified by the Tehama County Sheriff’s Office as Kevin Janson Neal, killed his wife, shot neighbors, attacked an elementary school and drove through the small rural community, firing at motorists Tuesday morning. He killed a total of five people and injured 10, including seven children, before he was killed by the police. One child who was shot at the school was in critical condition Wednesday; other children suffered less serious injuries from flying debris.

The shooting rampage lasted around 25 minutes and took place at different locations across the unincorporated community of Rancho Tehama Reserve, Phil Johnston, assistant sheriff of Tehama County, told reporters Wednesday.

Johnston said investigators believed that the rampage started with the killing of Neal’s wife, possibly Monday night. The police believe that Neal shot and killed his wife, cut a hole in the floor of their house, and placed her body inside.

“I really don’t know what his motive was,’’ Johnston said. “I think he had a desire to kill as many people as he could.’’

Neal’s sister, Sheridan Orr, told The Associated Press that her brother had a history of mental illness.

Police said the school’s decision to order a lockdown prevented many more children from being shot. A statement by the Corning Elementary School District said the Rancho Tehama Elementary School “was able to go on lockdown very quickly and effectively, which prevented any further injury or violence.’’

Johnston said that investigators had reviewed video from the school’s security system that showed the gunman walking the hallways and entering a restroom, but appearing to get frustrated that the classroom doors were locked.

The school went on lockdown at the sound of gunfire, Johnston said.

Johnston said that before the shooting, the sheriff’s office had received complaints from neighbors that Neal would fire weapons at his home.

Neal was under surveillance by police, but they had never witnessed previous incidents of gunfire.

Neal was arrested in January and charged with stabbing a neighbor. He posted $160,000 bail, according to Gregg Cohen, the Tehama district attorney.

As part of a restraining order, Neal was prohibited from having firearms. But the semi-automatic weapons he used in Tuesday’s attack “were manufactured by him, in his home,’’ Johnston said.