Except in most things, Judaism is patrilineal. Names are patrilineal. Tribe affiliations--including being a Cohen or a Levi--are patrilineal. I'm not an expert in Halacha, but I think it'd be fair to say that inheritance practices are more patrilineal than matrilineal. As far as I'm aware, only in one very specific case--the question of "Who is a Jew?"--does Judaism rely on matrilineal descent.

I actually tried to figure this out a while ago, and despite a lot of digging, I didn't come up with any clear answer.

The first question to ask is when did Judaism become matrilineal. Various offshoot sects including the Karaites, Samaritans and Beta Israel of Ethiopia are all patrilineal. So at very least there wasn't a strong concept of matrilineal descent when these groups split off (2nd temple period).

This makes sense because the whole idea of a religion that was separate from a tribal allegiance didn't exist until very late. In fact, I would argue that the main impetus for this new paradigm comes from the arrival of Christianity which broke the boundaries between religion and nationality, although you do see some of it earlier like in the story of Ruth.

Another question to ask is how does Judaism justify matrilineal descent. This is a bit different from the question of why the religion is passed down through the mother. Quoting myself:

How did Rabbinic Judaism justify matrilineal descent? The Talmud (Kiddushim 68b) justifies it from the verses in Deuteronomy (Deut 7:3-4):

3 neither shalt thou make marriages with them: thy daughter thou shalt not give unto his son, nor his daughter shalt thou take unto thy son. 4 For he will turn away thy son from following Me, that they may serve other gods; so will the anger of the LORD be kindled against you, and He will destroy thee quickly.

If you read this you might wonder why this seems to source matrilineal descent. The answer is that the second verse refers only to the son being "turned away" by God and nothing about the Jewish daughter being turned away. The implication, that the Talmudic rabbis draw, is that the daughter and her family remains Jewish, while the son and his family does not.

If you think like me, then this looks more like a post-hoc justification rather than an actual rationale. So it still leaves the why question open.

Again quoting myself:

If my historical reconstruction is correct, at the end of the second temple period religion and tribal allegiance were separated. Then various Jewish sects decided whether Jewish religion would be passed down from the father or the mother. Most sects chose the father, Rabbinic Judaism chose the mother. But why?

To be honest, I am not sure, and this is a question that I'm going to have to leave open. I am not sure why the society would favor matrilineal descent. It is very possible that the Talmudic exegesis quoted above was honest, and they came to this conclusion through textual analysis. It's also possible that they already had the custom and justified it with this verse. It's likely that they could have justified patrilineal descent through similar exegesis, as the other sects of Judaism did.

Tosafot (medieval commentaries on the Talmud) on Tractate Yevamos 45b states that if a person has a Jewish father but no Jewish mother, they are considered 'pagum' - flawed - but still considered Jewish.

Nowadays, Orthodox Jews believe patrilineal descent does not confer Jewishness (but Liberal/Reform Jews disagree).

So sometime in the past 500 years patrilineal descent went from being debated in Orthodox circles, to being meaningless.

Shaye Cohen in his The Beginnings of Jewishness has a chapter devoted to this. He determines that it would have taken place sometime in the second century CE and the two most likely options are the Roman and Jewish notions of mixing of seeds being conflated, such that inter-"race" breeding is akin to inter-species breeding. This makes sense because it's the same logic used for other things in the mishnah, like forbidding the mating of a horse and donkey, and when a horse and donkey breed they attribute the species to whatever is female. The second option he finds possible is a parallel to Roman practice that penalized a woman for marrying a slave or non-citizen by giving the child the status of the lower class partner rather than the mother's higher status. The result of this kind of illicit union in Judaism produces a mamzer, someone who is still counted as a Jew but has a lower status.

I don't find either explanation terribly compelling. There are a half dozen more options given in the chapter.