I think this is a problem for most writers in general given that there are no living truly matriarchal cultures if ever there were any.Though some evidence seems to suggest the Amazons were real. However, even in this case, their true nature has been obscured by centuries of myth and speculation and slander by their enemies. The only true matriarchies that exist only appear in the animal world at present, but these are not human societies so only a limited amount of information might be gleaned from them about how a matriarchal human/humanoid society might function.
Perhaps the best known and fully described matriarchy in fiction is that of the Drow in the Forgotten Realms and closely tied role playing game Dungeons and dragons. While it gives us an idea of how one might work, there is a substantial problem. It is largely portrayed as nigh irredeemably evil. Now while I feel that any gender based social control system is inherently evil, things are seldom so cut and dry. For example, in viking society, much inheritance, political power, and military might was controlled by men. However it was considered a grave offense to strike a woman (at least a viking woman) even in jest, women could own property, they could divorce their husbands if necessary, and on occasion joined up with the men in war. Women were also believed to be privy to certain magical powers that men did not have and the only other deity allowed to sit in their chief God's throne according to myth was his wife Frigga. I will not call Viking society a perfect or egalitarian one. I would be a moron selectively ignoring facts if I did. However this illustrates my point when I say such things are not always cut and dry.
Still there are societies I would call violently misogynistic which could be compared to the Drow based on the level of brutality and cruelty leveled against their fellow human beings in general, though much of their worst tendencies are directed upon women. The catalog of these offenses is so terrible and vast that I feel a certain rage rising up within me just thinking about it. I may be a man, but rage against injustice is something that crosses over gender lines and transcends them all together. The scale of the oppression of women that remains despite over a century of women's rights movements the world over is appalling. Any at all is truly appalling, but when one thinks about how much still remains, it is especially so.
Men can indeed be cruel, but so can women. From what I have observed, the capability for cruelty in either sex is equal, however the way this cruelty manifests is dependent on a number of cultural factors, what opportunities exist which are effected by cultural factors, and the person in question. Certainly we can say that Caligula was cruel, but we can also say Elizabeth Bathory was cruel. It seems to me that cruelty is dependent on social status and is almost seen as a sort of privilege for those of "higher" status (both Caligula and Bathory were high born people of privilege)! Which it should not be seen as no matter who holds the reigns of power if anyone at all.
All talk of the world's evils and the horrors of misogyny aside, the patriarchal system seems to stand on a particular institution of patrilineality. To clarify, patrilineality is the social institution by which people trace their lineage down their father's lines which for much of history not only included surnames, but material inheritance, titles, and status. From this we can infer that matrilineality would be a catalyst for a matriarchal society. Though neither always leads to one gender power structure all the time, it is likely that a matriarchal society would also be matrilineal.
What we also need to remember as I have been saying, is that these things are not cut and dry and exist on a spectrum. This is true of many things in human society from economics,religion, and government. For example, let's say we have a scale of monarchy which ranges from the least strict end with a constitutional government to the strictest form in which the monarch has nigh absolute power. We could also say this for monotheism. Let's say you have Christianity which believes in the strictest sense of the term that there is a singular deity of one gender (though some monotheists only use pronouns as a matter of linguistic convenience) while Hindus believe in a singular Godhead with many forms male and female. This can also be said of gender roles and gender based hierarchies in our world.
Personally when I write, one thing I hold to strictly is that in any culture there are noble people male and female. People who hold strictly to these structures and those who actively work to undermine them. That humans in general are capable of terrible things not matter what ethnic group, gender identity group, or religion they belong to. Therefore while we she should not be to willing to hold a matriarchal societies in fiction up on pedestals, we should also be careful not to demonize them in their entirety or make them seem inhumanly evil. There is a reason cultural relativism is a driving force in the study of society regardless of how we as individuals may feel about them.
So my only question is whether there is any additional advice one could use, or anything I forgot to mention as a result of this deities awful headache I've been having for the past few days.