Matriarchy is all about dictating/pretending to reach for equality of outcome. It pretends to tear down authority by asserting communal authority, ultimately an attempt to enslave everyone. It is Marxist style communalism. It is "abolition" of property.

Patriarchy is all about maintaining equality of opportunity to dictate. It asserts its goals of hierarchical authority more openly, attempting to enslave everyone. It is fascism or corporatocracy. It is assertion of entitlements/societal conditions to be "property."

Neither is the right answer. Both are a lie, and both are really the same thing with different rhetoric. Masculine and feminine forms of social dominance hierarchies. But the idea that any person has full authority over any other is simply erroneous, and leads you to be a slave to the madness.

Libertarianism is about equality of permission to do unharmful things and live your best life, including helping others to live theirs.