This question already has an answer here:
A girl is killed by magic in a wizard's household. Said girl never attended Hogwarts and there's nothing known about her, except that her mother, Kendra, also died in unclear circumstances not long ago. (Which case probably also wasn't investigated else Ariana would have taken to st. Mungo?) Doesn't a death caused by magic merits an investigation, casting priori incantatem on the wands present at the time of the death, collecting their memories in pensives to track back the events?