See my post #209. I was very deliberately using the term "matrilineal" to describe societies where the father has little-or-no confidence that any particular woman's children are biologically his, and children do not have a person holding a stable role of "father" in their lives (something like the situation in today's inner cities).
"Matriarchal" implies a society where women rule. Women do not rule in matrilineal societies like Somalia of South-Central LA -- nobody rules, there are just battles for dominance by various warring gangs
The distinction is important, because in societies where the children belong to the mothers (Africa, some parts of India, Native America, ghettos) men still "rule" in the sense that crime is out of control, rape is out of control, and war or war equivalent is prevalent. There is no society anywhere, in the present or in the past, where women "rule" in the sense that men rule under patriarchy.