Matthew's is the most Jewish of all the gospels. He maintains Christianity's continuity with its past with scholars such as W.D Davies thinking the most plausible reason why Matthew wrote his gospel was as a 'Christian answer to Jamina.' An example of the hostility to the Christians from the Pharisees is found in the birkit ha minim, "let them not be written with the righteous." So it could be argued that Matthew was writing to encourage his readers to keep going but also to define and defend his Christian community. Therefore when Matthew narrates an important event in the life of Jesus, he calls the reader's attention to a prophecy in the Old Testament of which this event is the fulfilment. Matthew's is the only gospel in which Jesus talks explicitly about the church. He wants to emphasise that the church did not begin after the resurrection, but was instituted by Jesus himself.
On the other hand, Matthew's purpose could be to present Jesus as the long expected Messiah. He is the king who has been sent to rule his people. There is from start to finish a royal aspect to this Gospel. At the beginning, Jesus' birth is announced, "Behold, a virgin hall conceive and bear a son and his name shall be called Emmanuel," which translates to 'God is with us'. So already, even before Jesus is born, Matthew attaches God to Jesus' status as the Messiah. Furthermore, Jesus' final words begin, "Lo, I am with you always, to the close of the age." This suggests that the role of the Shekinah of God is with Matthew's community.
Moreover, Matthew links his gospel to the Old Testament, quite frequently showing links to Moses. He seems rather found of arranging his material in numerical patterns so some think that Matthew arranged the main teachings of Jesus into five blocks to correspond to the five books of Moses. Also as far as Matthew is concerned, the Old Testament laws are still in effect. Not even an "iota or dot shall pass from the law," implying that not even the punctuation of the La is to be abolished. The Sermon on the Mount shows Jesus as a Moses-like lawgiver: tightening not abolishing the Law. When Marcion in the second century tried to sever Christianity from its Hebrew origins and to disown the Old Testament altogether, the Gospel he used was Luke's. After all, Luke had been written for the Gentiles. But Matthew shows us that we could not have the New Testament if it were not for the Old Testament and that Jesus would not be Jesus if it were not for the God of Abraham, Isaac, and Jacob.
Matthew's impression of Jesus is that he is of royal roots, possesses divine authority, and has the privileges and powers of God. The Matthean motive in the presentation of Jesus is his inherent relationship to the Old Testament, showing he is not a contradiction to Judaism but rather the climax and completion of Judaic faith.
Matthew seems to have various purposes. But, it most probably was written to demonstrate that Jesus was the Messiah. In this regard, the genealogy was probably meant to demonstrate Jesus' Messiahship to a Jewish audience that required proof of Jesus’ heredity; the miracles of Jesus affirmed Jesus’ authority as a spokesman for God; and the Old Testament links and quotations showed that Jesus is the fulfilment of the hope of Israel. Some motifs and themes in the Gospel indicate that Matthew wrote to encourage his audience to stand firm in their faith in their Messiah and resist the Jewish authorities who were persecuting them.