Who was the intended audience for the Book of Matthew?

One option is that it was written for the Jews of the Diaspora. These are still Jewish people, but few of them spoke or used Hebrew.

Because Matthew's Gospel, more than any other, relies on quotations and apparent prophecies from the Old Testament, some believe that the author was a Jew who wrote his gospel for other Jews. However, the Gospel was written in Greek and the Old Testament references were clearly taken from the Septuagint, an ancient Greek translation of the Hebrew Bible. For this reason alone, he must have been writing for a Greek-speaking audience.

When the author of Matthew cited the Septuagint, he accepted it as an accurate translation of the Hebrew Bible and was unaware of a number of errors in the Septuagint. Any Jew familiar with the Bible in its original Hebrew, would have quickly realised that Matthew was very much in error regarding the Hebrew Bible and could not have taken this gospel seriously. If the author of Matthew was writing for a Jewish audience, it would have been a Greek-speaking Jewish community that relied on the Septuagint and had little or no knowledge of Hebrew.

It could also be argued that it was written for illiterate Palestinian Jews who, following the destruction of the Temple, had no one they trusted to compare Matthew's Old Testament references with the Hebrew scriptures and would not listen to the teachings of the rabbis. A problem with this last argument is that the gospel was not written in Aramaic or Hebrew and would have been incomprehensible to most Palestinian Jews.

Bishop Spong says that the author knew little about Jewish religious laws and customs. He points out that in Matthew's nativity account, the presentation and purification stories are hopelessly muddled. If the author was writing for Jews, they would have become quickly aware of these problems.

It would seem more likely that Matthew was written for gentiles, and very likely for those gentiles known as 'God-fearers'. God-fearers were pagans who also worshipped the Jewish God.

Even if Matthew's gospel account was written initially in Hebrew, there are no autographs of any of the books of the Bible, so we could never know. The oldest available manuscript copies of Matthew are all in Greek. But that does not prove he wrote for Greek-speaking Jews, or Greek-speaking Gentiles. Greek was the language of commerce, it was used for all legal matters, and politics. Perhaps an illustration could be that in Europe there are many different languages used, but to facilitate commercial, legal and political and educational matters, English is the common language used throughout Europe. So in the first century; koine Greek was written.

We are therefore no further forward as to Matthew's intended audience until we start examining what he wrote. He starts with a genealogy that is entirely comprised of Jewish names, starting with the patriarch of Israel, Abraham, and including kings of Israel. Clearly, Jewish people would be keenly interested in such a connection. One of the opening verses gives another clue. The angel said (of the naming of Mary's child), "You shall call his name Jesus, for he shall save his people from their sins" (1:21): 'his people' being the Jewish people. So, in the first chapter, the attention of Jewish readers is particularly 'signaled'.

It is not until chapter 12 that Matthew makes any direct reference (in the sense of 'appeal') to non-Jewish readers. He does this by quoting a Hebrew prophecy about Messiah's name being what the nations (the Gentiles) will put their hope in (12:21 quoting Isaiah 42:1-4).

However, Matthew is unique, in all the gospel accounts, in mentioning the word 'ecclesia' - Church. He uses that word three times, yet it is never used in any of the other three gospels. This surely indicates that those people who knew themselves to belong to the Church that Jesus builds (Mat. 16:18) would see significance in this. By the time Matthew wrote his gospel, the initial preponderance of Jewish members of the ecclesia had moved over to a Gentile one. The word is used twice more in 18:17. Here Jesus is giving instructions on handling problems between members of the Church. Surely Matthew's unique mention of the Church shows that he was aiming his gospel account at those where were members of it? And those members were, by then, a mixture of Jews and Gentiles.

Supporting this, Matthew concludes his account by depicting the risen Christ appearing beyond the pale of Israel, looking out upon all nations from the heights of that appointed mountain in Galilee of the Gentiles, there to command his disciples. Those already members of his Church are told to "go and make disciples of all nations", showing that the Church was inclusive of Gentiles.

So, although the opening chapter begins with the Jews, the last chapter has the Gentiles included. And, throughout, the account is instruction to all who are members of the Church Christ is building, both Jews and Gentiles.

Matthew was likely written for Greek-speaking Christians of Jewish descent.

In my answer to your first question, I discussed reasons for believing Matthew was written in Greek rather than Hebrew. And it should be self-evident that the gospels were written to Christians. So here I'm going to list the reasons for believing the intended audience was specifically Christians of Jewish descent.

Scriptural fulfillment

The Gospel of Matthew, more than the other gospels, often concludes a passage by introducing a Jewish Scripture and claiming Jesus has now fulfilled it.

In many cases, Matthew is simply drawing a parallel between events from the life of Jesus and events from Israel's history. Examples are Matthew 1:22-23 quoting a passage about a sign given to King Ahaz, Matthew 2:15 quoting a passage about the exodus from Egypt, and Matthew 12:17-21 quoting a passage about God's servant Israel (see Isaiah 41:8-9 for identification of the servant with Israel).

Structure of the Gospel

Just as Matthew draws parallels between Jesus' experiences and those of Israel, the gospel itself parallels the Torah, the five books of Moses. In Matthew, Jesus' teachings are combined into five long discourses.

• Sermon on the Mount - Matthew 5-7
• Missionary Discourse - Matthew 10
• Parabolic Discourse - Matthew 13
• Discourse on the Church - Matthew 18
• Olivet Discourse - Matthew 24

Each of these discourses contains more uninterrupted teaching than the parallel sections of Mark or Luke.

Respect for Jewish law and customs

The phrase "kingdom of God" appears 62 times in the New Testament, but Matthew uses the substitute phrase "kingdom of heaven" 31 times, even where the parallel accounts in other gospels say "kingdom of God". This is likely due to a Jewish custom of never writing the word "God" as a precaution against using God's name in vain.

In Jesus' teaching about handwashing, Mark 7:19 indicates that Jesus thereby "declared all foods clean".

He said to them, "Then do you also fail to understand? Do you not see that whatever goes into a person from outside cannot defile, since it enters, not the heart but the stomach, and goes out into the sewer?" (Thus he declared all foods clean.)

But in the parallel in Matthew 15, this parenthetical comment is omitted. We know from Paul's letters that some Jewish Christians kept part of their unique dietary restrictions as a matter of conscience. Matthew perhaps did not want to offend their sensibilities.

In Mark 10 Jesus prohibits all divorce. However, in the parallel in Matthew 19, Jesus makes an exception for marital unfaithfulness. The version found in Matthew acknowledges that divorce is allowed in Deuteronomy under certain circumstances.

Furthermore, Jesus says in Matthew 5:17, in a statement not found in the other gospels, that he has come not to abolish the law but to fulfill it. (Fulfill is probably used here in the same sense as above.)

Conclusion

Matthew's attention to the Jewish scriptures and respect for Jewish customs, as well as the structure of this gospel, indicate that it was written primarily for Jewish Christians rather than pagan converts.