In a matriarchal way.

A female leader of a family, a tribe or an ethnic or religious group. / The dominant female in a family group of elephants

Governed by (or as if by) a matriarch. / Governed by females, rather than by males.

plural of matriarch

A matriarchal system or community. / The position of a matriarch.

The quality of being matriarchal.

matriarchal

matriarchally

plural of matriarchate

Synonym of matriarchy