matriarchy vs matriarchate

matriarchy and matriarchate both are nouns.

Word	Noun	Adjective	Verb	Adverb
matriarchy	Yes	No	No	No
matriarchate	Yes	No	No	No

As nouns, matriarchy and matriarchate are synonyms defined as:

matriarchy and matriarchate: a form of social organization in which a female is the family head and title is traced through the female line

matriarchy (noun)	matriarchate (noun)
a form of social organization in which a female is the family head and title is traced through the female line	a form of social organization in which a female is the family head and title is traced through the female line