Mark 11:27-33 (ESV)
And they came again to Jerusalem. And as he was walking in the temple, the chief priests and the scribes and the elders came to him,  and they said to him, "By what authority are you doing these things, or who gave you this authority to do them?"  Jesus said to them, "I will ask you one question; answer me, and I will tell you by what authority I do these things.  Was the baptism of John from heaven or from man? Answer me."  And they discussed it with one another, saying, "If we say, 'From heaven,' he will say, 'Why then did you not believe him?'  But shall we say, 'From man'?"— they were afraid of the people, for they all held that John really was a prophet.  So they answered Jesus, "We do not know." And Jesus said to them, "Neither will I tell you by what authority I do these things."
Questions of authority, these are usually the wrong questions to be asking. They don’t ask Jesus why he is doing what he does. They ask by what authority. Obviously he hadn’t done it with the approval of the High Priest. One could probably guess he didn’t have the governor’s approval either, or Herod’s. But what authority did he need to drive robbers from the temple? What authority does one need for that kind of thing? Of course Jesus had the authority, it was his father’s house. He had the authority from heaven, he was the authority. But he decides to put the Pharisees to their own test before answering. He asks them about the baptism of John, it puts them in a bind. If they answer it was from heaven, not only will he ask, why then did you not believe him? But it validates his own ministry. If they answer from man, they will lose face with the people who believed John was a prophet.
So Jesus doesn’t tell them. His time is coming he knows it. He isn’t trying to avoid the cross, but they are going to have to work a bit harder to get him. And they won’t get him until he decides the time is right.