That Jesus is the son of David is a consistent theme throughout the book of Matthew. In 21:9 Jesus is called the Son of David as he enters Jerusalem, a clear messianic title at that time. When asked, the Pharisees respond (accurately) that the Messiah is the Son of David. Reading Matthew from the beginning, we know that Jesus is the son of David (Matthew 1:1, 1:17). But the Pharisees refuse to accept that Jesus is the son of David!
To follow up their answer, Jesus quotes Psalm 110:1, an important Psalm which implies the Messiah will be the son of David. Patriarchal society gives the father ultimate respect, so one would expect the son of David to be subservient to David. But that is not the case in Psalm 110. Jesus points out that David is speaking “in the Spirit”, or under inspiration. He is not speaking falsely, he is telling the truth, that the Messiah, his son, is above him as Lord.
Psalm 110 is an important messianic psalm in the New Testament. Who is the first Lord in Psalm 110? The word is Yahweh, the second lord is Adonai, In the original context, this might be God saying to David, I have anointed you as king; alternatively, it could refer to Solomon, the son of David who is seated alongside David on his through, anointed with oil, brought into Jerusalem on a donkey, and inaugurates a kingdom of peace and prosperity. Sitting at the right hand, alluding to enthronement at the ascension of Jesus.
How can the Messiah be a “son of David” and a high priest since he is not from the tribe of Levi? He is a priest in the order of Melchizedek. Melchizedek was a king of Salem and priest of the Most High God who blessed Abraham (Genesis 14). Since he is appears in Genesis without any introduction fascinating some later writers and he became a messianic figure (see this post in Melchizedek in Hebrews 7). Melchizedek was the king of Jerusalem in the age prior to the Davidic Kingdom. Since David captured Jerusalem as his capitol, he became a king in the line of Melchizedek. As the son of David, Jesus is both a king (in the line of David) and a priest (in the line of Melchizedek).
This son of David will put “enemies will be under his feet.” This cannot refer to Solomon, although he experienced peace and prosperity (foreshadowing the future kingdom). The apocalyptic imagery of 110:6, judging the nations and heaping up the dead. This cannot refer to the reign of Solomon, David, or any other king of Israel or Judah. Philippians 2:5-11 echoes this, since Jesus is obedient to the father, he is exalted to the very highest place (the right hand of the father is not mentioned, but he is seated), and given a name that is above every name.
Jesus’s response to the Pharisees goes back to the original question in Matthew 21:23-27: Where does the authority of the Son of David come from? This returns to the question in Matthew 21:23-27, where did Jesus get his authority? Jesus demonstrates his authority throughout Matthew, by the way he teaches, through his miracles, etc. Now he asks the Pharisees to give an answer: where that authority comes from?
Pharisees are speechless with this question, and they do not dare to ask him any further questions. They likely have enough to convict him of heresy, they would not want the crowds to become any more convinced of Jesus’ claims.
The conflict with the Jewish leadership is over and they have been silenced. They are now afraid to ask Jesus any questions! Jesus then turns to his disciples and the crowds and condemns the Jewish leadership (and the Pharisees in particular) for their utter hypocrisy as leaders of Israel.