Mary's answer is plausible.
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One approah is that the reason is Historical. "Judaism introduced matrilineal descent somewhere in the period of the Second Temple and the Roman occupation according to Prof. Shaye Cohen, now at Harvard..."
"I recall that it was less the issue of being certain of the father's identity than it was to bring Jewish law into somewhat coherence with Roman law regarding "citizenship" and "nationality" and rights and privileges within society..."
( http://judaism.about.com/library/3_askrabbi_c/bl_matri_descent.htm and see also http://www.myjewishlearning.com/ask_the_expert/at/AsktheExpert--Matrilineal_descent.shtml )
"This practice may have originated in the period of Ezra (Ezra 10 3; Neh. 13.23 ff) and may parallel that of Pericles of Athens who sought to limit citizenship to descendants of Athenian mothers. It may also have represented temporary, emergency legislation of that period."
( http://data.ccarnet.org/cgi-bin/respdisp.pl?file=38&year=carr )
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There are also those who proclaim that Jewish ID was always passed through the mother, even in Biblical periods. They say that there are allusions to matrilineal descent in Tanakh. ("This began with the giving of the Torah and is rooted in the law in Deuteronomy." - http://www.askmoses.com/en/article/174,2063982/Why-is-a-childs-Judaism-determined-by-the-mother.html )
"Leviticus 24:10 speaks of the son of an Israelite woman and an Egyptian man as being "among the community of Israel" (i.e., a Jew).
On the other hand, in Ezra 10:2-3, the Jews returning to Israel vowed to put aside their non-Jewish wives and the children born to those wives. They could not have put aside those children if those children were Jews."
( http://www.jewishvirtuallibrary.org/jsource/Judaism/whojew1.html and http://www.jewfaq.org/whoisjew.htm#Matrilineal )
"The Mishnah (Kiddushin 3:12) states that, to be a Jew, one must be either the child of a Jewish mother or a convert to Judaism. The Talmud (Kiddushin 68b) derives this law from the Torah. The relevant Torah passage (Deut. 7:3-4) reads: "Thy daughter thou shalt not give to his son, nor shalt thou take his daughter to thy son. For they will turn away thy son from following me, that they may serve other gods." "
( http://en.wikipedia.org/wiki/Matrilineality#Judaism )
See also http://www.faqs.org/faqs/judaism/FAQ/05-Worship/section-35.html and
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"Matrilineal descent, although generally accepted for the union of a Jew and a non-Jew, has rested on an uncertain basis. Some have deduced it from Deuteronomy 7.4, others from Ezra 9 and 10. Still others feel that the dominant influence of the mother during the formative years accounted for this principle. A few modern scholars felt that the rabbinic statement followed the Roman Paulus (Digest 2.4 f), who stated that the maternity was always known while paternity was doubtful; this, however, could be extended to the offspring of any parents. Shaye Cohen has also suggested that the rabbis may have abhorred this type of mixture of people as they felt negatively toward mixtures of animals and materials. A full discussion of this and other material may be found in Aptowtizer's "Spuren des Matriarchats im jüdischen Schriftum," Hebrew Union College Annual, Vols. 4 and 5 and Shaye J. D. Cohen's "The Origin of the Matrilineal Principle in Rabbinic Law," Judaism, Winter, 1984."
( http://data.ccarnet.org/cgi-bin/respdisp.pl?file=38&year=carr )
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There are some who give "spiritual" reasons for this law. For example, see http://www.israelnationalnews.com/Articles/Article.aspx/9005 and http://www.chabad.org/library/article_cdo/aid/601092/jewish/Why-is-Jewishness-matrilineal.htm But I don't think that's what you were looking for.
(See also http://lhaim.blogspot.com/2006/11/why-is-being-jewish-determined-by.html )
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See http://www.oztorah.com/2009/07/matrilineality-is-still-best-for-jewish-identity/ and http://en.wikipedia.org/wiki/Matrilineality_in_Judaism for a discussion of this issue.
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There are two academic works that touch on this topic:
"Who Was a Jew?: Rabbinic and Halakhic Perspectives on the Jewish Christian Schism" by Lawrence H. Schiffman
"The Beginnings of Jewishness: Boundaries, Varieties, Uncertainties" by Shaye Cohen