Patriarchy, if the term means anything, is patrilineal, meaning that the family line is based on the father. The sons inherit, the daughter gets a dowry to get married.
The way to achieve this traditionally was monogamy, which coincidentally also makes the father part of a child's life, bringing great advantages for the child.
Matriarchy, on the other hand, is matrilineal, meaning that nobody knows or cares who the father is, with great disadvantages for the children. Unlike the father, the mother is always known. Polygamy is the norm here.
Men come and go while the mother has the inherited resources and the children aways stay with her.
Right now we are in the transition from patriarchy to matriarchy, leading to a whore epidemic, which is nothing but women competing for the men with the most resources instead of getting a man who corresponds to them.
It's not yet a full matriarchy and maybe never will be, but polygamy is already normalized and has created a vicious competition.
The result is a catriarchy, which means that a lot of women remain unmarried because they didn't win the competition and now only their cats have an ongoing family line.