Molins, the Paris prosecutor, also praised Beltrame, who "at the risk of his life took the choice to take the place of the hostages".
Lieutenant Colonel Arnaud Beltrame, who once served in Iraq, was taken to hospital after the siege in which he took the place of a female hostage at the Super U store in the town of Trebes, near the Pyrenees mountains that divide France and Spain.
Prime Minister Edouard Philippe called the incident "a terrorist act", and the Paris prosecutors' office said an anti-terrorism investigation had been opened.
The attacker began his spree of attacks on Friday morning in Carcassonne, where he first hijacked a auto, killing one passenger - whose body was later found hidden in a bush - and injuring the driver.
Then Lakdim shot at four police officers jogging in the same town, injuring at least one.
The hostage-taker was now alone with one police officer in the supermarket and all other hostages had been freed, he told BFM.
Hundreds of police officers were deployed to the area which was cordoned off.
Carole, who was shopping at the supermarket, described how people had taken refuge in a cold room.
Neighbours said the young man who lived with his parents and three sisters seemed "calm" and "nice". "So I went to see what was happening and I saw a man lying on the floor and another person, very agitated, who had a gun in one hand and a knife in the other one".
"We were 10 people and we got out by the emergency exit at the back".
Most of the hostages managed to escape but Lakdim kept one woman back as a human shield.
France's top security official says a heroic officer who swapped himself for a hostage during an attack on a supermarket has died of his injuries.
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Interior Minister Gerard Collomb confirmed on Friday afternoon the gunman had been shot dead.
When police heard gunshots on the phone line, an elite Swat team stormed the supermarket.
Nearly six years ago to the day, Islamist gunman Mohammed Merah killed seven people in the Toulouse region, about 90 km from Carcassonne. The officer was also seriously wounded.
What do we know about the suspect?
"He was known by the police for petty crimes, we had monitored him and did not think he had been radicalised", Mr Collomb said after arriving at the scene of the hostage-taking in the town of Trebes, near Carcassonne.
He had received a suspended prison sentence in 2011 for carrying a forbidden weapon and a one-month sentence in 2015 for consuming drugs and obstructing police. Salafism is a hardline offshoot of Sunni Islam.
The family's apartment was raided by police on Friday afternoon.
What has the reaction been?
The ISIS-affiliated Amaq News Agency said that the gunman was a "soldier" of the militant group and that he had responded to its call to attack coalition countries, a reference to the USA -led coalition fighting ISIS in Syria and Iraq.
By all appearances, Lakdim was motivated to carry out his attacks by extremist Islam.
Col Beltrame was grievously injured, and his death raises the toll from the attack to four.
Soldiers continue to patrol major tourist sites around France and transport hubs under an anti-terror mission. Last month, he emerged from almost two years of isolation in a French prison to be tried in Brussels over the shootout that resulted in his capture.