Doesn't it make more sense that the lineage to David's house would be on Mary's side?

Father Maletta Responds: Your question is very observant and from our world view make good sense. However, in the world that Jesus was born into, one’s only identity was from one’s father. Matthew’s Gospel was written to a Jewish audience who would not be able to culturally comprehend an identity generated from the maternal. Furthermore, Matthew is trying to show his Jewish readers that this faith in Jesus is not a new religion but a fulfillment of Judaism. To do this Matthew regularly demonstrates that Jesus was foretold by the Jewish prophets, that the facts of Jesus’ life match these ancient Jewish prophecies. In Luke’s Gospel the lineage looks different because Luke is writing to a gentile audience so the lineage is traced back to Adam to show that Jesus’s gift of salvation is intended to all people. Matthew is appealing to Jews to believe in Jesus while Luke appeals to gentiles; Matthew presents Jesus as the fulfillment of the Jewish law and prophets, Luke as the Christ for all people. Not unlike tailoring a message for a certain demographic the two gospel writers shape their presentation to make the biggest impact on their targeted audience. The culture of that time was far more symbolic than we are today so when Matthew claims Jesus is from the house of David he is making a powerful connection to this symbolism. Legally, culturally and practically Jesus was the son of Joseph – if only by adoption – so it is appropriate for Matthew to make the Davidic claim.