“house.” The magi were not at the birth of Christ. They came over a year later. Joseph and Mary were in a “house,” but the Scripture does not tell us whether they were staying with someone else or had their own house by then. Bethlehem was a small town, and the magi would have had no trouble finding the right house. No one would have forgotten what the shepherds had said less then two years earlier when Jesus was born, how angels appeared to them and said the promised Messiah had been born.
“child.” The Greek is paidion (#3813 παιδίον), which means “young child.” Jesus was no longer a “baby,” which is the Greek brephos (#1025 βρέφος), as he was in Luke 2:12, 16. Now, at over 1 year old, he is a young boy.
“with Mary his mother.” The Magi no doubt saw Joseph too, but Matthew’s focus is on the Messiah himself, and the woman who was divinely impregnated by God. Thus the record mentions Jesus and Mary, but not Joseph.
“paid homage.” See commentary on Matthew 2:2.