I don’t care about any law but the Law.
You mean the Law of Moses?
I don't recall anything in the Pentateuch that says a Jew is someone born of a Jewish mother. (In fact, they were called Israelites rather than Jews back then.)
The Old Testament is fairly patriarchal, whereas the rule you cite is matriarchal. I suspect it originated long after the Mosaic law.
Where does this rule come from? The Talmud? Some other Jewish tradition? When does this rule first appear in writing? The medieval period? Later still?