Does your Mother suffer from Alzheimers? If so, her strange behaviour may be a symptom. If, however, she is in full possession of her faculties, and someone genuinely has ransacked her house, taken her credit card, and walked out with a bagful of loot, I would suggest that the Police should be involved, as a crime has clearly been committed. If that wasn't bad enough, your Mother thinks that you are the criminal. A Police investigation will provide an opportunity for them to quiz the neighbour and to, in effect, call her bluff. The friend with the key will also be questioned. This is the only way in which the truth is likely to emerge. Sorry you're going through this, and I do hope that you can clear it up quickly.