Continuity mistake: When Jack and his lover go to an elegant restaurant on their first date, she removes her coat with nothing covering her black dress, but when they are seated at their table, she is wearing a white cover across her shoulders. She appeared to be empty-handed when she arrived.
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 Mathilde is ordering Jack the rifle at the bar, the position of her newspaper and glasses on the table changes between shots.
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 Jack gives Mathilde the rifle at the bar, the number 4 plastic sign on the table keeps moving around 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.