The Gospel of Matthew has several incredibly strong arguments that link it firmly to the New Testament and show without much doubt at all that is should be definitely included in the New Testament. Firstly, it corresponds very closely with two other gospel accounts of the life of Jesus, so much so that Matthew, Mark and Luke are called the synoptic gospels, referring to the "common" story that they share and tell about Jesus and his life and ministry. This offers significant proof that it is accurate and reliable, as it account tallies with the account given in other books that are recognised as being part of the New Testament.
Secondly, it has overt and deliberate ties to the Old Testament, and it seeks throughout its pages to show that Jesus is the fulfilment of all of the Old Testament prophecies and promises. Matthew is the gospel that contains most verses from the Old Testament, and it also, in Matthew 1, seeks to establish a direct link from Abraham to the birth of Jesus, as Matthew 1:17 demonstrates:
Thus there were fourteen generations in all from Abraham to David, fourteen from David to the exile to Babylon, and fourteen from the exile to the Christ.
The genealogy that begins the gospel thus presents Jesus as a direct product and outcome of the Old Testament. Lastly, the Gospel of Matthew should be included in the New Testament because it is the gospel that was used most by the early church, and it was cited by the early church fathers more than twice as often as other gospels. This indicates very strongly that the early church immensely valued this gospel, and therefore church tradition supports its inclusion in the New Testament.