As stated elsewhere, we only have tradition that the apostle Matthew wrote the Gospel that now bears his name. A careful reading in the original Greek language shows that Matthew takes over almost all Mark's material, Mark's sequence of events and, for the most part, Mark’s wording. Uta Ranke-Heinemann puts the consensus of New Testament scholars succinctly in Putting Away Childish Things, when she says, at page 218, that it is incomprehensible that an eyewitness (the Apostle Matthew) would choose to depend so radically on a non-eyewitness (the author of Mark). The real author of Matthew is unknown.
Mark's Gospel mentions both Levi and Matthew separately, referring to Levi, son of Alphaeus as a tax collector whom Jesus called to follow him (Mark 2:14). Mark never again refers to Levi, who is not mentioned in the full list of the twelve disciples (Mark 3:14-19), where Mark introduces other disciples including Matthew, Thaddaeus and James, son of Alphaeus. So far, there is good reason to see Levi and Matthew as being two separate people. Luke follows Mark closely, mentioning Levi but only in the context of a story in which Jesus is criticised for consorting with tax collectors (Luke 5:27-32), with Matthew being one of the twelve (Luke 6:15).
Disciples are not meant to change their minds when called by Jesus, yet this seems to happen when Mark omits Levi in the list of all the twelve apostles. Matthew's author resolves Levi's unexplained absence simply by not mentioning Levi at all, and by having Matthew as the disciple who was a tax collector, so that two thousand years of tradition have held that Levi and Matthew must be the same person.
We have two sources (Mark and Luke) that say Matthew and Levi are two different people, and one source that does not mention Levi, even if Matthew transfers the description of Levi onto the person Matthew. On this evidence, we should be compelled to see Levi and Matthew the apostle as two persons. The evidence that the author of Matthew's Gospel could not have been Matthew himself, means that we have three separate people here.