I'm looking for a manga that have matriarch family or clan leader. where the matriarch-ness is deep looted and traditional. I'm NOT looking for a hentai.

can you be a bit more specific with what your asking for? i cant really understand what you want, do you want something like Kekkaishi where they stick to the family code of protecting there area/fighting youkai or more along the lines of naruto??

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You mean a clan or community traditionally lead by a woman... so not Naruto! The closest I can think of is Otoyomegatari in that there are matriarchal figures carrying on the traditions.... but the society definitely isn't matriarchal. Sorry I can't be of more help.

In Kisshou Tennyo the protagonist's family is said to have been preserving a maternal lineage, allowing an eldest daughter to succeed the household.

yah kekkaishi definitely counts. tokkin comes from a matriarch family with matriarch tridition. Naruto don't count. just because the leader is female doesn't make her a matriarch, since there is no tridition of female leaders.

Takeru-sword of susanoh-Its not in the beginning of the manga but about the end of volume 1 it introduces a city of Amazon women that have female soldiers and the throne passes from mother to daughter.

In Good witch of the west the throne passes from mother to daughter, there have always been queens and the queen regulates the world.

Operation liberate men-the protagonist is taken to a parallel world which is ruled by women, gender roles are reversed and the throne passes to female heirs. So instead of the typical wishing for male heirs its female heirs that are desired and men are typically slaves in that world too. Though the women aren't very motherly, the atypical attitudes of males and females are also mostly reversed.

Marginal has a system where there is only one female and they are the holy mother. They kind of rule, but are more a figure of worship.

Ruri- the mc comes from an island village/estate which is passed to female descendants. Though to be honest only the first part of the manga is about this, afterwards the mc and the girl she is competing for the island with get transported to a fantasy world. The thing about the island is only relevant to understand the relationship between the two girls.

Amatsuki-It has this supernatural buddhist temple that is passed down to 'princesses' but they're androgynous, so not sure if it fits XD.

Bokura no Kiseki-Argh I can't remember exactly what it is that they inherit from each other, (its bothering me, hence the argh) but I know that there is something that only the daughters of the royal family can obtain....or at least rituals that they go through, if that helps...

Gekka bijin-I think this is a matriarch society, but I really don't remember if it's come about because there are only female heirs or if its supposed to be like that.

If the transfer of knowledge or power counts then:

Legend of Chun Hyang has shaman/healing/magic skills passed mother to daughter.

Witches have a couple of stories of inheriting power or magic if you're a female and others where the female was stronger using magic/shaman powers.

This might fit but there's not much translated yet.
Four Daughters of Armian

oO that's totally ok, i'm korean lol