Could it loosely be traced back to a matriarchal system in the very old days in these regions?

Some comparative-historical linguists have theorized that Proto-Indo-European did not have a tripartite (masculine, feminine, neuter) grammatical gender system, but rather a dual one (animate, inanimate). It seems that the feminine gender developed from a reanalysis of the plural ending of inanimate nouns as an abstract and singular one: cf. Latin bonum 'good thing' and bona (earlier, literally 'good things', but later 'goods'; Greek also tends to use singular forms of verbs with neuter plural subjects. The German linguist Johannes Schmidt (link) did some of the early work in his Die Pluralbildungen der indogermanischen Neutra (1889, link).

Ceci n'est pas un seing.