A fact that some do not whish to acknowledge is that it is not only who rules the society or has the most power but also the social structure of the society that defines matriarchal and patriarchal.So at best a feminist whishes to be one in the patriarchal setting. One particular reasoning is that marriage is a Christian invention and Christianity is obviously patriarchal.If you want to keep everything in a patriarchal society the same but change who has the most power than only the roles have changed sides. I understand equal rights but I don't think anyone else realizes that they want it in a patriarchal system.
It's not entirely true that marriage is a Christian invention. Marriage predates recorded history, which puts it well before the rise of Christianity, to many thousands and thousands of years ago, and it has existed in various forms in various societies all over the world for a very long time, including parts of the world that never saw a monotheistic religion of any kind for several thousands of years.
"Equality" does not mean that everyone is the same, or even that everyone has the same power, for whatever that is worth. Clearly, everyone is not the same. Equality means "equality under the law" ... that is, no one is denied the protections and privileges of the law simply because they're female or because their skin is black or because they're gay or because they're children or because they talk to God in their own way. "Equal rights" has nothing to do with patriarchal or matriarchal. It simply means that the same law applies to everyone.
And besides, this just sounds like more crap to try and rationalize injustice. It won't work. You don't have a rational leg to stand on.
Interestingly enough, all the evidence shows that the "partriarchal" society that humanity built for itself was born out of equality, not some sort of male superiority or dominance. At any rate, it's a thing of the past.