i) A conventional objection to the traditional authorship of Matthew is that an apostle wouldn't make use of a secondhand source like Mark. There are several problems with that objection:
ii) According to Acts 12:12, Jerusalem was Mark's home town. So Mark may well have had firsthand knowledge of Jesus whenever Jesus came to Jerusalem.
iii) Moreover, Mark hits many of the major points in the life of Christ. It's not as if Matthew is going to omit those events. Since these are key events in the life of Christ, we'd expect him to repeat them. And there's a rough chronology to the events, so why would Matthew make a special effort to change the plot?
iv) But here's another factor that's overlooked. Mark's mother hosted Christian gatherings in her home. Peter knew the location of her home. Indeed, the slavegirl knew the sound of his voice (v13), so he must have been a frequent visitor. But then, it must have been known to the other apostles. It stands to reason that Mark had many opportunities to befriend other apostles, as long as he and they were in Jerusalem.
So what if Matthew was one of his informants? If Mark writes about some things he didn't personally observe, what was his source of information? Given his access to some of the apostles, they'd be a prime candidates.
Indeed, he may have gotten information from several apostles, but if only two of them wrote Gospels, that's our only basis of comparison. We wouldn't recognize the input from other apostles who never penned Gospels.
Suppose he questioned Matthew about Jesus, and incorporated that into his Gospel. Later, Matthew reads Mark and thinks to himself, "Well, as long as Mark is using my material, I might as well write up my own recollections to include additional material that I didn't mention to Mark." Or something like that.
Would Matthew be using Mark? It might appear that way, given the order in which they were published. But the Apostle Matthew can one of Mark's sources even though his Gospel was published after Mark's Gospel.
In that case, Matthew isn't using Mark; rather, Mark is using Matthew. It's just that Mark published some of Matthew's material before Matthew got around to publishing his own material. To some degree, it was Matthew's material all long. Mark borrowed from Matthew before wrote his own Gospel. Indeed, Mark's Gospel may have given Matthew the stimulus to do his own.
Incidentally, if Papias is right, it's possible that Mark made use of some catechetical material that Matthew originally produced in Aramaic.
v) If that sounds convoluted, here's a comparison. Wayne Grudem is one of John Frame's students. Grudem published a popular systematic theology.
Over 20 years ago, Frame mentioned in class that reading Grudem's Systematic Theology was a bit of a deja vu experience because he noticed that Grudem had incorporated some of Frame's lecture material into his systematic theology. Years later, Frame began turning more of his own classroom lectures into hefty books.
Now, a keen-eyed reader who compared the two, reading them horizontally, might be struck by parallels between Grudem and Frame. Since Grudem wrote before Frame, he might conclude that Frame borrowed from Grudem. But it's really the other way around. You can't infer the order of conceptual dependence from the order of publication. Grudem borrowed from Frame, not vice versa.
vi) Incidentally, this can be a cause of bitter feuds in the history of math and science. The question of priority. A scientist or mathematician may have been the first person to discover something or formulate a theory. And he scribbled it down. But he didn't publish it right away. Sometimes he's scooped by another scientist or mathematician who got it published first. Sometimes that's an independent development, but sometimes the published scientist or mathematician got it from the unpublished scientist or mathematician in private conversation or private correspondence. Watch the fur fly when he steals his thunder.
So I'm not making some outlandish proposal. This is a pretty commonplace distinction, both in principle and practice.