With this said, the US history could be way different. Women did not vote back in the 1600's-1800's, so the women could have voted for someone else. Making a new president. Unlike George Washington as are first president.
Households would also be different. In this picture, Alan is with his family. His wife, Sarah is in charge of raising the kid, and Alan spends little time with his daughter. Alan also lives in a different house and comes over occasionally if Sarah wants him to.
One part of society that would be the same is that people will still own houses and get money. If there was none of that that would cause a major change in this universe. There would still be big cities in the country.
Another thing that would be the same in a matrilineal America is that America would still be a country. There would still be founding fathers and Daughters( that just sounds better, And mothers too). The matrilineal society has men as political leaders, so the America today would look similar but not the same as today.
The final thing that would be similar in this society to modern America is that all People will have a role in the country. The roles would be a little different, but all people will have roles. Which shows that it is similar.
But, I think that the America we have today is great and that his Matrilineal society thing will not work. These examples are not giant counties like the US, but small society's, where this best works. USA! USA! USA!