Matthew 1:1-17 begins with Abraham and ends with "And Jacob begat Joseph the husband of Mary, of whom was born Jesus, who is called Christ. Notice that it does not say Joseph begat Jesus; it says that He was born of Mary. Jesus was "begat," but not of Joseph; He was the only begotten Son of God, Himself (Jn. 3:16). The reason Matthew began his genealogy with Abraham is because his Gospel was written to verify that Jesus was Israel's long-awaited Messiah, their King, very God manifest in the flesh (Matt. 1:1; 14:33; 16:16; 27:37; 1 Tim. 3:16).
Luke 3:23-38 is not only in reverse order to Matthew's, it presents Jesus in His humanity, being both the Son of Adam, and the Son of God (v. 38). To be certain that no one believed Jesus was the Son of Joseph, Luke's genealogy begins with "And Jesus Himself began to be about thirty years of age, being (as was supposed) the son of Joseph" (v. 23). In one of the strangest prophecies of the Bible, God said that Eve's offspring would be the One who would eventually crush Satan: "And I will put enmity between thee and the woman, and between thy seed and her seed; (He) shall bruise thy head, and thou shalt bruise His heel (Gen. 3:15). Mary did not become pregnant by Joseph's seed, but "her seed."
Some have pointed out that there is an error in Matthew's genealogy. Matthew wrote, "So all the generations from Abraham to David are fourteen generations; and from David until the carrying away into Babylon are fourteen generations; and from the carrying away into Babylon unto Christ are fourteen generations" (v. 17). When you count the names listed, there are only forty-one individuals, when three times fourteen is forty-two. But notice, Jechonias was listed twice, before the carrying away of Israel (v. 11), and after their "carrying away" (v. 12). Why Matthew saw fit to do this, I have no clue. It really is a moot point in that it is obvious that every generation in the genealogy of Jesus was not included. Jesus was the seventy-seventh generation from Adam (see www.ldolphin.org/2adams.html).
Another amazing difference between the two genealogies is that one mentions women, and the other one doesn't. You would think that Matthew, being written to show Jesus was the Jewish Messiah, would be the one to avoid listing women, since to the Jews, women played no role in their heritage of a man. But believe it or not, it is Matthew that mentions Thamar (v. 3 Tamar), Rachab (v. 5 Rahab), Bathsheba (v. 6 "the wife of Urias"), and Mary (v. 16). Mary is not mentioned by name in Luke's genealogy, and yet it is the genealogy of Jesus through the lineage of Mary!
If the Gospels were fiction, wouldn't the two genealogies be the same?