"Matthew and Luke record contradictory genealogies for Jesus. Mathew says Jacob is Joseph's father and Luke says Heli is his father (Matthew 1:1-15 vs Luke 3:23-38)."

The genealogies don't contradict each other because Matthew and Luke intended to trace different lineages. Matthew recorded Jesus' legal lineage through Joseph, while Luke recorded Jesus' biological lineage through Mary. However, this raises an obvious question:

"Why would Luke call Joseph Heli's son?" 

Joseph was considered Heli's son in a legal sense when he married Heli's daughter (Mary), according to the law of Moses. If Heli only had daughters, which seems to be the case since the Bible only mentions Mary and her sister (John 19:25), then there was no way to directly continue his lineage through a firstborn son. But according to Mosaic law, his daughter's husband would receive the inheritance and the daughter would become part of her husband's tribe, merging their lineages (Numbers 27:1-11, Numbers 36:1-12). That way, Heli's lineage could continue through Joseph as his 'son', while his daughter and grandchildren could still receive the family inheritance. Although this seems odd to us today, it wouldn't have for a first century Jew under Mosaic law and patrilineal descent. 

"Why wouldn't Luke simply include Mary into Jesus' lineage? After all, isn't it supposed to be her side of the family?"

Patriarchies only trace lineages through men. This means Luke would have never listed Mary, even though he was showing Jesus' lineage through her side of the family. Instead, Joseph legally represented her as a link in the chain from Jesus to Adam. This is consistent with the biblical idea of a husband's headship over his wife (Numbers 36:1, Ephesians 5:22-32). They became "one flesh" and he represented her in the lineage.

The opening of the Gospel of Matthew in the  Liesborn Gospels . Germany, 10th century A.D.

The opening of the Gospel of Matthew in the Liesborn Gospels. Germany, 10th century A.D.

"But it would be unusual for people in a patriarchal society to even bother sharing someone's lineage through their mother's side." 

This is absolutely true. However, the virgin birth was a unique situation and it makes sense why Luke would want to include it. Since Matthew's account only shows a legal connection to king David for his Jewish audience, it was important for Luke to show Jesus' biological connection to David. That way, we can see that Jesus met the requirement to be messiah both legally and biologically. 

"Matthew's account includes women, why wouldn't Luke's?"

Matthew did mention women but he didn't include them in the patrilineal descent. He merely mentioned significant wives of the men Joseph descended from. We don't know why Matthew does this and not Luke, but since Luke doesn't mention women at all, it makes sense that Mary wouldn't be mentioned either. She's only indirectly mentioned in Luke 3:23. 

 

 

"How do you know Matthew and Luke weren't conspirators trying to deceive their readers or wrote contradictory genealogies out of ignorance?"

The explanation of dual lineages is much more plausible than alternative explanations. It's clear that Matthew and Luke weren't even trying to trace the same lineage since they differ on every link in the chain from Jesus to David. They even differ on the identity of Joseph's "father", which would have been easy to figure out since Jewish lineages were very public and accessible for tax and inheritance purposes (Numbers 36:1-4, Luke 2:1-5). And even with a late dating of the Gospels (90AD) many of Jesus' contemporaries would have still been alive to identify the alleged error. If Matthew and Luke were attempting to match genealogies but got them wrong, we would expect some similarities between the two; they would start off agreeing on Jesus' grandfather, then veer off after several generations with contradicting names. 

Furthermore, Christians throughout church history didn't see this as a contradiction. If skeptics are right that the genealogies and Bible are fabricated by deceptive Christians, then the early Christian editors would have corrected this obvious "contradiction". However, early Christians didn't see it as a contradiction and have offered several much more plausible explanations throughout history. Alternative explanations of conspiracy or ignorance simply don't hold up.