Why did the mid-early church consider the Gospel of Matthew to be reliable?

The Gospel of Matthew is now considered to be important, however the 1st century and 2nd century references to it are surprisingly confusing and sparse. Which authors in the 1st and 2nd century refer to it *unambiguously*? For each author, please note the following: 1. When they wrote. 2. Whether they cite the Gospel of Matthew's content. 3. Whether they refer to it as the Gospel of Matthew. 4. Whether they seem to have considered that it was written by Matthew the apostle. 5. What criteria they use for acceptance as canonical. **Examples** * Tertullian seems to have written ~190AD (?), but doesn't associate the contents of what is now the Gospel of Matthew with Matthew the person, and instead associated it with Luke. * Clement (end of 1st century?) seems to quote Matthew, but does not say it was written by Matthew, or indeed give it a name or authorship of any sort. * Papias (150AD) mentions a Gospel of Matthew, but gives the impression that it was written in Hebrew and does not cite it, so doesn't leave us certain that he was talking about the same one. * Iranaeus (190AD) gives what is most similar to the modern account of the authorship of the Gospel of Matthew. (I assume that he cites it so that we can be sure that he saw what we see in the gospel of Matthew?) So: Are the above correct? Are there other 1st and 2nd century sources that refer to and/or cite the Gospel of Matthew and its authorship? What criteria did they use for determining authorship and canonicity?