The Gospel of Matthew is anonymous: the author is not named within the text, and the superscription "according to Matthew" was added some time in the second century.
The tradition that the author was the disciple Matthew begins with the early Christian bishop Papias of Hierapolis (c. AD 60–163), who is cited by the Church historian Eusebius (AD 260–340), as follows: "Matthew collected the oracles (logia: sayings of or about Jesus) in the Hebrew language (Hebraïdi dialektōi), and each one interpreted (hērmēneusen – perhaps "translated") them as best he could."

On the surface, this has been taken to imply that Matthew's Gospel itself was written in Hebrew or Aramaic by the apostle Matthew and later translated into Greek, but nowhere does the author claim to have been an eyewitness to events, and Matthew's Greek "reveals none of the telltale marks of a translation".
Scholars have put forward several theories to explain Papias: perhaps Matthew wrote two gospels, one, now lost, in Hebrew, the other our Greek version; or perhaps the logia was a collection of sayings rather than the gospel; or by dialektōi Papias may have meant that Matthew wrote in the Jewish style rather than in the Hebrew language.
The consensus is that Papias does not describe the Gospel of Matthew as we know it, and it is generally accepted that Matthew was written in Greek, not in Aramaic or Hebrew.
The Gospel of Matthew is anonymous: the author is not named within the text, and the superscription "according to Matthew" was added some time in the second century. The tradition that the author was the disciple Matthew begins with the early Christian bishop Papias of Hierapolis (c. AD 60–163), who is cited by the Church historian Eusebius (AD 260–340), as follows: "Matthew collected the oracles (logia: sayings of or about Jesus) in the Hebrew language (Hebraïdi dialektōi), and each one interpreted (hērmēneusen – perhaps "translated") them as best he could."
On the surface, this has been taken to imply that Matthew's Gospel itself was written in Hebrew or Aramaic by the apostle Matthew and later translated into Greek, but nowhere does the author claim to have been an eyewitness to events, and Matthew's Greek "reveals none of the telltale marks of a translation". Scholars have put forward several theories to explain Papias: perhaps Matthew wrote two gospels, one, now lost, in Hebrew, the other our Greek version; or perhaps the logia was a collection of sayings rather than the gospel; or by dialektōi Papias may have meant that Matthew wrote in the Jewish style rather than in the Hebrew language. The consensus is that Papias does not describe the Gospel of Matthew as we know it, and it is generally accepted that Matthew was written in Greek, not in Aramaic or Hebrew.
Matthew the Apostle