Early church writers appear to indicate that Matthew’s gospel was written first. Irenaeus [2nd Century AD – c. 202], in Against Heresies 3.1.1 [c. 180 AD], says “Matthew also published a gospel in writing among the Hebrews in their own language, while Peter & Paul were preaching the gospel and founding the church in Rome. But after their death, Mark, the disciple & interpreter of Peter, also transmitted to us in writing what Peter used to preach. And Luke, Paul’s associate, also set down in a book the gospel that Paul used to preach. Later, John, the Lord’s disciple — the one who lay on his lap — also set out the gospel while living at Ephesus in Asia Minor”.
This paragraph was taken from Wikipedia’s article on Markan Priority. It is found in the section labeled “Arguments against Markan priority.” (Bracketed dates are added, and taken from the linked articles.)