Continuity mistake: When Jack gives Mathilde the rifle at the bar, the number 4 plastic sign on the table keeps moving around between shots.
Continuity mistake: After Mathilde's gun explodes, the amount of blood on her face increases and the position of her hair changes from one frame to the other.
Continuity mistake: When Jack is driving to the river, from the front angles his knuckles are covered in blood, from the side angles they are not.
Continuity mistake: When Mathilde is ordering Jack the rifle at the bar, the position of her newspaper and glasses on the table changes between shots.
Answer: Because she didn't, but once she saw evidence of who/what he was, he had to kill her to protect his identity.