Xenophile, that's all very nice, but that'd be an equal society, not a matriarchy.
Though I have to admit, the differences aren't that big - women just don't have a need to oppress men. The matriarchy described in the linked article doesn't much appeal to me - why on earth would I want to have naked men on page 3?
BapsOfSteel, if I had a say in it, girls would do strength training at school. The female body needs more exercise to maintain muscle mass than the male body, and I'd want girls to be as strong as boys, or at least not at as much of a disadvantage. The boys could do dancing instead. Or take lessons on how to respect women's boundaries. (Okay, that would not be gender-specific, it would be sex specific)
Strength training, if done efficiently, allegedly is only necessary two times a week, so after school sportsy activities could be freely chosen.
Okay, not matriarchy, either, just equality. I suppose I, too, am just too nice to invent a matriarchy that does to men what men in real life to do women.
The only thing I can come up with, and I actually suspect that's just equality, too: No paternity rights for unmarried fathers, or married fathers who didn't do at least 50% of childcare for at least a year.
I mean, really, what does a man contribute to a pregnancy? Five minutes of fun - why should that entitle him to any rights?