What do you imagine to be the benefits of patriarchy and matriarchy?
The answer to this is really a matter of personal opinion. Our answers to this question will differ based on what we believe about the inherent natures of men and women. If we believe that men and women are inherently different, we will probably believe that both patriarchy and matriarchy have their own particular advantages. If, on the other hand, we believe that men and women are essentially the same, then we will not want either patriarchy or matriarchy. Instead, we will want a system in which neither sex dominates.
I, personally, do not believe that men and women have different natures. Therefore, I do not think that either matriarchy or patriarchy holds any benefits for a society. I think that societies are better off when they treat men and women equally and when both sexes share equally in the running of a household.
However, in order to help with your question, I will give some ideas as to what someone might think is good about patriarchy and matriarchy. People who believe that patriarchy is good probably believe that men are more qualified to work outside the home and that women are more suited to working within the home. If that is the case, one benefit of patriarchy is that children will be better cared for. Women will stay home with their children and will do a better job of raising them than men could. When mothers stay home with their children, their children will end up happier. The family will also be better off because the man will do a better job of providing for the family. Supporters of patriarchy might also imagine that a patriarchy would see more economic growth because they believe that men are more suited to economic activity.
People who believe that matriarchy is superior are more likely to believe that women are more naturally caring and unselfish. If you believe this, you will believe that a society dominated by women will be more just. Men in such a society will not be allowed to abuse women and children. Men will not be able to create a masculine, “dog-eat-dog” economy that results in great inequality. Men will not be able to bring about the sort of levels of violence and conflict that we see both within and between nations. In a matriarchal society, the women in charge would run things in such a way that everyone would be treated fairly. They would solve problems without resorting to macho posturing and competition.
These are benefits that some might see in patriarchies and matriarchies. I personally do not agree with either of these points of view, but I present the arguments as others might make them.