Have there been studies comparing patriarchial and matriarchial societies?

The most significant and scholarly text on the matter is possibly Steven Goldberg's “The Inevitability of Patriarchy”. His conclusions, as far as I can summarize them here, were three-fold.
1- Patriarchy was ubiquitous in all known historical societies (with a few exceptions), even among the more nominally matrilineal ones.
2- “Patriarchs” inevitably expressed their powers on “ancillary” matters pertaining to outgroup interactions- with other tribes & with deities, whereas “Matriarchs” had significant & even total power over ingroup dynamics.
3- The exceptions Goldberg observed were inevitably two-fold. One- mythical societies being considered from an external perspective, such as the Classical Bharatas alternately despising & lusting for the women of Stridesha, and twonear Stone-Age barbarians struggling to invent the wheel. Even among the latter, the men lived near-independent, violent, taboo-ridden lifestyles which wouldn't have been out of place with early IE Berserker and Wildman motifs.
Readers are encouraged to study the text itself, if they can, since it's obvious that I'm writing from memory and might be mistaken on minor points.
Lastly- I disagree with Goldberg on one major point.
While Goldberg - one of the great sociologists & anthropologists of his time - might've been accurate on most points, I personally feel he neglected to consider the Cyclicity of History & Future trends of postmodern societal development as well as he could have. Goldberg saw little sign that the Patriarchical Industrial society would change in any significant manner in the future. This is clearly false.
Furthermore, Goldberg considered societal dynamics as mostly static, and thus ignores how the nature of a Society as a Patriarchy or a Matriarchy might reflect its “Virtue”- or self-sufficiency and resilience. However that is possibly explainable as such topics do not come under the ambit of his studies.

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