The Islamic State group's Amaq news agency claimed Chekatt was a "soldier" of the group but Interior Minister Christophe Castaner rejected the claim as "totally opportunistic".
People began returning to the area on December 13 with many marking their respects for the victims by leaving candles in the main Kleber square.
Security forces, including the elite Raid squad, spent hours Thursday searching in the Neudorf neighborhood where Chekatt had grown up based on "supposition only" he might have been hiding in a building nearby, a French police official said.
Castaner said authorities chose to reopen the market "for the honour ofStrasbourg, for the honour of France".
A local police official, who spoke on condition of anonymity because he was not authorized to speak publicly, said the man who shot at police Thursday night had been armed with a pistol and a knife.
Interior Minister Christophe Castaner said three police tried to question Chekatt after spotting him "wandering" through the streets in the Neudorf area of the northeastern French city where he grew up, but he opened fire.
"We want to reconstruct the past 48 hours in order to find out whether he got some support", Mr Heitz said.
The Strasbourg attack was the latest in a succession of attacks linked to the Islamic extremism in France going back to March 2012.
Access to the market has been reduced while extra police and military troops have been deployed as well as private security guards.
Questions remain over how Chekatt was able to evade the tight security perimeter set up around the Christmas market, which has always been a prime target for militant groups. "It's part of our common events and belongs to all the French people", Castaner said. "And this morning we just wanted to show my stay here that we are able not to bow" - he said during a visit to the scene of the attack.
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"Our engagement against terrorism is total", French president Emmanuel Macron, who was in Brussels for a European Union summit, said in a tweet thanking security forces.
He said seven people were in custody, including four of Chekatt's family members and three in his "close entourage" - two of them detained Thursday night.
He turned to fire on them and they shot and killed him, Castaner told reporters.
"A projectile hit the vehicle above the left rear door, two police officers responded, shooting several times, and killed him", Heitz said.
Police are now focusing their investigation on whether Chekatt had any help in carrying out his attack or while on the run.
More officers arrive at the scene soon after, followed by crime scene investigators who take photos of the body and the surroundings.
The office of the Paris prosecutor, which handles terrorism cases in France, later confirmed it was Chekatt. Chekatt was a Strasbourg-born man with a lengthy criminal record who also had been identified as a security risk.
Witnesses said the gunman shouted "God is great!" in Arabic and sprayed gunfire from a security zone near the Christmas market on Tuesday. He had his first conviction at 13, and had 26 more by the time he died at 29.
Strasbourg's location in the heart of western Europe means that Germany, Switzerland, Belgium and Luxembourg can be easily reached by auto or train, making the search for Chekatt more complicated.