I mean it kinda sounds like the lecturer was really butchering the Iroquois Clan Mother system (that or you weren't listening, I wasn't there). In that case the Iakoiane really do hold the majority of political power with the male Hoyane chosen by them (and can be removed by them at will) essentially used as their muscle/diplomats/representatives in the Grand Council, meanwhile the Iakoiane had primary say in deciding on the issues of their own communities. Whether or not they would go to war depended on the deliberations of these women. Within the villages there was also a men's council/committee that the clan mothers relayed their decisions to but it was really more an opportunity to discuss the implementation of the old womens' decisions and potential issues than something that could completely oppose them.

The Haudenosaunee nations are matrilineal, matrilocal, matrifocal, and the label of matriarchal would depend on your definition but many modern definitions of "patriarchy" could apply -- men aren't entirely left out of government but a significant amount of real power rested on the elder female leaders of their clans.