Matrilineal Descent
Matrilineal descent is learned indirectly from the verse that follows the commandment not to marry a non-Jew. This is based on the Talmud in Masechet Kiddushin.
The wording of the Torah is not clear and needs explanation. The Torah speaks of a situation when, “Your son will be removed from Me, and will worship false gods.”
Rashi explains that this verse is speaking to the grandfather, the father of the son who has turned away from Hashem. The proximity of this verse to intermarriage implies that the son has turned away to such an extent that he has married a forbidden woman.
The result of this marriage might be that she gives birth to a boy. This grandson is not your son, but is considered her son.
The Torah did not speak of a case where your daughter turns away and worships idols, because the son that she might bear, is considered her son, and your grandson.
It would seem to be a lot easier if this was explained more openly, but this is the tradition that began with Moshe Rabbeinu, and is known as Halacha Le’Moshe Mi’Sinai.
This term, “Halacha Le’Moshe Mi’Sinai,” means that it is a rule that is etched in stone. It came as instructions directly from Hashem to Moshe. We have no authority to question laws of this category, just as we are not allowed to question G-d. Laws of matrilineal descent fall into this category.