Deep down Matthew is torn by his decision. I mean his mother is almost certainly dead, he had not seen his father or heard his father since the morning, so he is most likely dead. The Vault has been his home for 18 years, he has known nothing else. Leaving the Vault was a last resort option for Matthew. So there is definitely a part of him that does not want to leave the Vault, and he'll surely try to get back in once he thinks it is safe.