[Graymatter]

Matrilineal societies are not "dominated" by women, they just have a different perspective on relatives and how they're classified.
Not all societies "trade" females. I don't have time to go into it, but for example, in some societies, he goes to live in HER house, and their children's closest male relative isn't him, it's her brother. (And the brother is the "dominant" one. The bridegroom can dominate over at his sister's house.)
It's not easy for us to follow, even in the basics, never mind the particulars. But there are still such societies, in Asia, and up into the 1800's I believe, there were a few in Asia Minor (islands along the coast).
Go to the local library and look for a basic textbook on cultural anthropology. Some of the kinship systems people have used---successfully---can make our eyes crossed, but they exist even today.

[muawiyah]

Even chimpanzees trade females~!

[Question_Assumptions]

I'm aware of the distinction. I'm also aware that it tends to get overblown and overinterpreted into many things it isn't. Otherwise, there wouldn't be books like Cynthia Eller's The Myth of Matriarchal Prehistory: Why an Invented Past Won't Give Women a Future. It would be fair to say that this particular case could be a harmless interpretation and could be correct but I see a red flag whenever I see an interpretation like this.