A Matriarch is a significant female within a family or group, sometimes taking a leadership role or holding some cultural significance within the group.
The Last-of-all-Cities colony was led by Matriarch T'Kell. (EV comic: "Cloak and Dagger")
The Ilaiyenai of planet Sigma Niobe II were led by Matriarch Damala. Most other Niobean cultures were also matriarchal. (TOS short story: "As Others See Us")
In the Pa'haquel species, the wife of the Clan elder served as Clan matriarch. If the Elder was unmarried, the position fell to the most senior female member of the Clan Elder's line. In 2380, the matriarch of Clan Qui'Tir'leg was Qui'chiri. The duties of the matriarch included preparing newly killed star-jellies for habitation. Deanna Troi told Qui'chiri that she considered the position of first officer to be comparable to that of matriarch. (TTN novel: Orion's Hounds)
Though males like D'Lor were responsible for representing the House of Mo'Kai in the chambers of the Klingon High Council, it was common knowledge that the true power inside the clan rested with its matriarchs such as J'Ula, who ruled from behind their puppets. These female leaders were also responsible for overseeing various secret experiments that the Mo'Kai nurtured, including the ChoH'a', which was first tested on the albino Voq after his lover L'Rell — a member of the aforementioned house — sent him to her matriarchs to undergo the surgical procedure in order to infiltrate the Federation Starfleet and claim victory in the current war. (DSC - The Light of Kahless comic: "Issue 3"; DSC episode: "The Butcher's Knife Cares Not for the Lamb's Cry")
Among the Holy Order of the Kinshaya, the Matriarchs and Patriarchs were part of the ruling Episcopate. Females in the priestly class were restricted to civil government.
Following the Niamlar Massacre on Janalwa in 2382, the Matriarchs issued public statements reasserting the divine infalibility of the Pontifex, and in the face of calls for a transition to a more populist government, Pontifex Ykredna and the Matriarchs and Patriarchs vowed to resist such efforts.(ST - Typhon Pact eBook: The Struggle Within)
- The exact duties and responsibilities of the Matriarchs are unclear.