The Jews today, or the Talmudic Jews, basically use these scriptural references to support the Matrilineal authenticity of Jewish identity: If the mother is a Jew, then the children will be Jews.
Deuteronomy 7:1-5, is used to infer such: "your child would be turned away by "HE-the gentile" is assumed as the child's Jewish state in trouble.
Leviticus 24:10, The son of an Israelite woman and an Egyptian man was among the community of Israel-a Jew.
Ezra 10:2-3, Putting away strange wives and their children.
(Source: http://www.jewfaq.org/whoisjew.htm)
Obviously, the dispensationalists are easily bought into this error as well.
And making matters more complicated than God's intended, ie. grandchildren identification, etc.
2 Corinthians 3:6 warns us how not to misinterprete God's words. On top of Romans 9-11 regarding Israel's identity. (Such as some would think I support the matrilineal claim even if they have read this entry because I did not specifically state that a child born of a Jewish mother may not be Jewish.)
Jesus said: For whosoever shall do the will of my Father which is in heaven, the same is my brother, and sister, and mother. (Matthew 12:50)