I was wondering as a Christian, how sure are we that Matthew the Disciple of Jesus wrote the Gospel that bears his name and if he did, when can we assert he wrote his Gospel?
I wish I could say that we know for sure that Matthew wrote this gospel and that we know the date it was written. Unfortunately, both statements would not be true. What we have is the clear tradition in the church from the earliest possible date that Matthew wrote the book. This tradition goes all the way back to Papias in about AD 130. We have Matthew quoted by Clement by the very late 1st century AD, so we know it was written and already considered "scripture" in the late first century. Most reliable conservative scholars will put the date of writing of Matthew in the 50s or 60s AD, although some will even move it into the 40s. However, we do not have the autograph (actual original) of the gospel and we will have to be content with knowing that the tradition is unanimous that the apostle Matthew wrote the gospel. The majority of scholars believe the original was in Greek, not Aramaic which was presumably Matthew’s first language. Whether or not Matthew was sufficiently versed in Greek to pen the book in that language or whether or not there was an original in Aramaic is an open question, but these are reasons some doubt Matthew wrote the book. As for the date of writing, given the mention of the destruction of Jerusalem, which happened in AD 70, is treated as a future event, and given that the rebellion which led to this destruction started in AD 66, I personally believe the book was written AD 65 or earlier.
Matthew entered the Christian canon of the New Testament because it was considered to have apostolic authority. In other words the apostles or those near to the apostles agreed the the book was authoritative. This in and of itself does not prove that Matthew wrote the book. Nearly all liberal scholars believe he did not, and even some conservative scholars say it is not likely he wrote the book, while others are more or less convinced he did. For us, what is sufficient is to know that the Gospel of Matthew was accepted by the church in the first century as authoritative and inspired.