Just an addition to the fine comments already posted on this subject.
Though no extant manuscripts exist of it, it appears that the Gospel account which we know as “Matthew” was originally just an oracle or “sayings” gospel–a collection of Jesus’ teachings and sayings without narrative–originally penned in Hebrew/Aramaic.
According to critical analysis, this source could likely be the original composition of St. Matthew, of whom Papias of Hierapolis wrote: “Matthew put the oracles [or “sayings” of Jesus] in an ordered arrangement in the Hebrew language.”
While it is clear that the narrative was written in koine Greek (and the order that Jesus’ teachings and sayings appear in Matthew show that the sayings were arranged into a very specific but non-sequential order) this is believed to be of a later date than the collection of sayings/oracles. The finalization of Matthew came about either by the hand of the original author or via the particular church group which St. Matthew presided over, using his teachings and recollections to provide the final arrangement.
There is now a manuscript transmission theory that presents this original “sayings” gospel by Matthew as the long-lost “Q” source. If “Q” really existed, it is believed that it would be such a collection of sayings by apostolic authorship (which fits the bill for Matthew’s “sayings” source). If this is true, then a long lasting debate might be ended between those who argue for Mark as the first gospel penned and those who agree with the traditional view that Matthew was penned first.
If “Q” is St. Matthew’s original sayings gospel, then it is agreed this was written before Mark. The “sayings” source is clearly older. Mark’s gospel was edited into its final form afterwards, with the redaction of Matthew into its final form (with the Greek written narrative and the oracles translated in Greek as well) completed afterwards. This would also explain why Luke uses “Q” as important source material, as it is a recollection of the sayings of Jesus by someone who was there to listen to them firsthand and in the language Jesus spoke (again presuming the theory holds).
This is theoretical of course, but it might answer some questions. Either way it is agreed that the “sayings” or “oracles” as they stand in Matthew had a Hebrew/Aramaic source, even though we don’t have that original source available.
The same is understood about Luke’s “Nativity Narrative” which seem to be sourced from Mary herself and show some signs of struggling as Luke attempts to put the terse Semitic expressions into the Greek tongue.