What would a true matriarchy look like in today's world?
Is that like asking a fish what it would be like to live in the air?
I guess to start to answer, you have to identify what are the features of patriarchy, and imagine the opposite - or is that too simplistic?
Taking an idea from fiction, it could be a society in which 'paternity' was just not an important thing. DNA tests for paternity would be never done, unless there were very specific medical reasons for needing to know genes.
Women would have babies with whoever, and if they had a regular male partner in their lives, he would assume a 'fatherly' role whether or not he was the father.