7 Now Luke tells us also the cause wherefore he proceeds to write: “that you may hold,” says he, “the certainty of the words wherein you have been instructed;” that is, that being continually reminded you may hold to the certainty, and abide in certainty.
But as to John, he has himself kept silence touching the cause; yet, (as a tradition says, which has come down to us from the first, even from the Fathers,) neither did he come to write without purpose; but forasmuch as it had been the care of the three to dwell upon the account of the dispensation, and the doctrines of the Godhead were near being left in silence, he, moved by Christ, then and not till then set himself to compose his Gospel. And this is manifest both from the history itself, and from the opening of his Gospel. For he does not begin like the rest from beneath, but from above, from the same point, at which he was aiming, and it was with a view to this that he composed the whole book. And not in the beginning only, but throughout all the Gospel, he is more lofty than the rest.
Of Matthew again it is said, that when those who from among the Jews had believed came to him, and besought him to leave to them in writing those same things, which he had spoken to them by word, he also composed his Gospel in the language of the Hebrews. And Mark too, in Egypt, is said to have done this self-same thing at the entreaty of the disciples.
For this cause then Matthew, as writing to Hebrews, sought to show nothing more, than that He was from Abraham, and David; but Luke, as discoursing to all in general, traces up the account higher, going on even to Adam. And the one begins with His generation, because nothing was so soothing to the Jew as to be told that Christ was the offspring of Abraham and David: the other does not so, but mentions many other things, and then proceeds to the genealogy.
Source: Homilies on the Gospel of St. Matthew (New Advent)