There are still a few cultures in the world, most of them in remote locations, that follow this type of inheritance and descent. In many ways, it makes a lot of sense. There is rarely any doubt who the mother of a child is. With all property handled by women, there is less difficulty in providing for the needs of the family. Unlike many modern cultures where rules have to be put in place to provide for widows and orphans, matrilineal societies more easily incorporate them into their hierarchy.
I wonder how different our world would be if this sort of society had maintained dominance into modern times.