It is well known in Jewish law that a child is Jewish if his mother is Jewish. This is known as "matrilineal descent." I will try to explain the basis for this law.
There are two places in the Torah that deal with the subject of matrilineal descent. The first relates to the subject of the Hebrew slave. A Jew can be sold into slavery if he was a thief and was unable to pay for the item he stole. As a form of rehabilitation, the court will assign him a Jewish family to live with and work for over a period of six years.
During those six years, his master may give him a non-Jewish slave to sire children with her. The Torah is very specific in telling us that at the end of six years, when the Hebrew slave goes free, "His (non-Jewish) wife and children belong to the master, and he goes free."
The children are considered her children and not his, and they also have the status of Canaanite slaves. This case is our first proof that the offspring of a Jewish man and non-Jewish woman, follow the mother.
The second proof comes from today's Parsha that discusses the prohibition of intermarriage in Chapter seven of Deuteronomy. This is explained in the Talmud in Tractate Kiddushin 68b.
The Torah is very specific by telling us, "You shall not give your daughter to his son, and you shall not take his daughter for your son." The next verse begins with the words, "When your son shall be removed from before Me and he shall worship other gods."
The Talmud explains that this verse is only referring to the case of your daughter marrying a non-Jewish man but not the case of your son marrying a non-Jewish woman. The reason being that the son that comes from such a union is "her son" but not your son.
The belief based on the Talmud is that this tradition taught my Moses himself was that the offspring of the union of a Jewish man with a non-Jewish woman is "her son" and not your son. Hence, that child is not considered Jewish according to Jewish law.
Hopefully, this gives a little better understanding of the origin of matrilineal descent. We must remember that the Jewish people managed to survive through a very harsh exile, due to their strict adherence to the Torah and traditions handed down from generation to generation.