The transition to matriarchy will destroy civilization for two reasons.
1. Civilization is indivisible from the nuclear family, and the nuclear family proceeds from patriarchy. Absent a leadership role in familial life, there is no incentive for male responsibility and thus dramatically lower male investment in society. Beyond this, males raised outside of nuclear families are consistently less psychologically prepared to become responsible, civilized men.
2. The scale of an advanced society proceeds from a male system of social organization, which revolves around sprawling, distended, and impersonal networks of mutual benefit rather than direct, personal, and human bonds. As the transition continues, our institutions will progressively scale down in size.
I am still unsure whether this will be a universal development or whether the economic bifurcation identified by Charles Murray will divide a civilized 1% from a post-apocalyptic majority. I do believe, however, that his social bifurcation is real and will occur partially along gender lines, creating a polygamous upper class and heavily male lower class.