When did matrilineal descent designating a person as a Jew begin? If it began in the tribe of Jacob, or after the Torah was given then can Avraham be considered Jewish? How do we know that Moses' wife and children are Jewish? The reform Jews claim that being Jewish initially was determined by patalineal descent. Is there any validity to this?

There are 2 covenants: that of the Forefathers, and that of Israel. The status of Israel as a nation began with the national exodus from Egypt and the subsequent giving of the Torah at Sinai, and culminates with the national entering into the Land of Israel. Accordingly, the rabbis describe Mount Sinai as a type of mass conversions (not literally but figuratively) and from then on we are the nation of Israel with the Torah of Israel. Beforehand, the sons of Israel (=Ya’akov), and even Moshe, married from outside the “tribe”, but we weren’t a nation yet, so it was allowed. The source that the lineage of Israel goes after the mother is from Dvarim 7, 3-4, “...you shall not give your daughter to his (=the gentile’s) son… for he will turn away your (grand)son from following Me”, where we see that your daughter’s son from a gentile man is nevertheless, considered "your son" and we don't want him led astray by his gentile father. This is also mentioned explicitly in Ezra, 10, 2-4, which goes back more than 2,500 years. Among the explanations, are because of the decisive influence of mothers’ upon their children, others explain the decisiveness of knowing she’s the mother (as opposed to the father, who’s identity is more questionable), while others explain that this is simply the Torah’s decision, whether we understand it or not.