This article touches on several different topics including: What language did Matthew originally write in, did he write primarily for a Jewish audience, was the Christian community understood as essentially distinct from the Jewish community at the time the Gospel was written, was the author of Jewish or Gentile descent, how/why did the author have very specific knowledge of Roman language and culture, and where was the Gospel originally written.
One consensus of the article’s different perspectives, backed by Patristic sources, is that the Gospel was not originally written in Greek, but in either Hebrew or Aramaic. Also, I agree with the more widely accepted belief that the Gospel was intended for an audience with a Jewish background (held by some of the Church Fathers), whether that is because they themselves are ethnically Jewish, or because they have a firm knowledge of the teachings of Judaism. The opinion that Matthew wrote the Gospel for a singular community (community theory) is far too narrow minded and un-probable. It seems to make more sense that Matthew surely knew of other Christian communities that were experiencing the same difficulties as the one (or ones) he was personally around, thus, he would have written the Gospel with the intention of being used by many Christian communities. Between the distinction of Jewish Christians (extra muros) or Christian Jews (intra muros) I lean towards the opinion of extra muros. Although his points on who is asking the question (e.g. a Pharisee or an Apostles) seems convincing, there does seem to be a very prevalent, essential difference between Jews and Jews who choose to follow Christ stretching throughout the Gospel (obviously not implying that Christ abolished the Law, but that He fulfills the Old Testament). Regarding the last topics, I think the best conclusion is that Matthew was a Christian of Jewish descent, who was well educated in both Jewish and Roman history and tradition, and who wrote the Gospel around the area of Syria, most likely in Antioch.