Why is the Jewish race determined through the mother instead of the father? What’s the symbolism here?

Shecaniah’s call for the Jews to put away not only their foreign wives but the children produced by marriage with these foreign wives, and that this be done “according to the Law” (Ezra 10:1-3), shows that hundreds of years after Moses established the Torah, the Jews’ understanding of the Law of Moses was that it taught that if the mother of a child is Jewish, that child is Jewish.  Conversely, if the mother is not Jewish, then the child is not Jewish even if the father is Jewish.  The Jews of Ezra’s day would not have put away children they considered to be Jewish.  Interestingly, there is no explicit command in the Torah specifying Jewish matrilineal descent, even though clearly the Jews of Ezra’s day considered such to be the case.  Yet the Code of Jewish Law clearly teaches such today, and rabbinic tradition going all the way back to biblical times has also taught that Jewishness has followed the mother.

I can only speculate as to the meaning behind it.  In my research I came across a Jewish source who said the purpose behind it was to keep the family line purely Jewish due to the common practice of the men having concubines and mistresses.  A child therefore would be considered Jewish only if a Jewish wife was pregnant.  The example of Hagar and Ishmael and Sarah and Isaac shows that the children of concubines and slaves were not considered heirs unless there was no child by the legal wife.  More study needs to be done as I’m sure there’s more to this question than the answer I’ve thus far provided.