Matthew, unlike Mark, focused the beginning of his account on the ancestry of Jesus, which is a primary concern for Jewish- Christian readers, hinting at his possible audience. The opening “title” βιβλος γενεσως (book of genealogy) corresponds to תולדת ספר in Gen. 2:4, 5:1 (The book of the genealogy of Adam, of Noah etc.). Matthew intends to demonstrate the genealogical legitimacy of Jesus of Nazareth to be the Messiah of Israel.
This interest in the genealogy suggest that Jewish concerns will be a primary interest in the Gospel. Therefore, Matthew builds on the concept of prophetic fulfillment in the stream of Jewish tradition. For example, Matthew's usage of the phrase: "ίνα πληρωθη" (that it might be fulfilled) through out his account, asserts a concentrated effort to the conclusive fact that Jesus is the long awaited Messiah.
Thus we can conclude that the purpose of Matthew's writing is to demonstrate to the reader that Jesus of Nazareth is truly the expected Messiah. He is qualified to be the King of the Jew by his family line and by the many instances in which he had fulfilled the words of the prophets.