10:12 AM On the original motivation for Matthew:
As soon as the first wave of converts had been baptized and their instruction organized by the Twelve, the apostles’ thoughts turned to the practical question of how to unify and consolidate their teaching about Jesus. The apostles realized that they somehow needed to promulgate those passages of the Holy Scriptures from “Moses and all the prophets” (Luke 24:27) that Jesus had explained to Cleopas on the road to Emmaus. It also became clear to them that their main apologetic task was to demonstrate to the Jewish authorities that Jesus had literally fulfilled all the prophecies about the Messiah. These considerations were the original motivation for the composition of the Gospel of Matthew.
From Why Four Gospels?