Matrilinearity

There are crimes which are caused by sheer jealousy. One of these is the case where a man kills his wife, and or his children, because the relationship has broken down and, as he sees it, the wife and children are no longer "HIS". He can't stand the idea that they might become the PROPERTY of some other man.

In UK society, which is the one I know best, and in many classes, there are single women bringing up their children because the man has left, or been made to leave. This is in itself almost a matrilineal society. The children may or may not have the name of their father, depending on whether the couple married or not; but marriage is becoming increasingly rare as it is commercialised to the point of idiocy and people are drawn to expect their parents to spend many thousands of pounds on what should be a simple public ceremony.
(I've met far too many women who longed to marry their sometimes decades-long partner, but simply couldn't afford the wedding.)

In a matrilineal society, the mother, left with the children, would receive some kind of support from her "line" - her mother, her relatives; a sister or aunt. She might move back to her mother's house.
Many of the single mothers I have known have been forced by circumstances, usually money or loneliness, to take a new partner very soon after their breakup.
(The fact that women's work is STILL far lower-paid than that of men may have something to do with this... also state benefits that are completely impossible to live on.)
This can lead to all kinds of further problems, where the new partner, a little like a lion, may resent the children of his predecessor. The children too, of course, may well resent the man they see as their father's replacement. Tensions follow, and the second relationship all too often breaks up.

My thought, I suppose, is that a matrilineal style of living might be a better one for many women, and for their children, and also that it is very difficult, in a culture that is overwhelmingly patriarchal, to achieve that goal. But that it might well be a direction we need to move in, if our culture is to survive at all.

(In case you were wondering, I am very happily married in RL and have been for 36 years. Marriage DOES work for some of us, and matrilinearity would not invalidate marriage for those who want or need a long-term partnership.)