A problem for getting power from power jack

I am a quite beginner in Arduino. I used the power jack to a supply source for my Arduino Mega and confirmed the input voltage was 8.8V. But the board seems could not work normally because the 5V power output and 3.3V power output were floating. After I inserted the USB cable to the board, it works, on board LED bright, power outputs became normal.

So, do you think that my board is broken?

From your description, the autopower switching does not seem to be working. Is your 8.8V source a DC source, and how many mA of current is it rated for?

Yes, it is a 9 V AC to DC adapter with 500mA. So, how can I confirm the autopower switching does not work?

What do you mean by floating? What did Vin measure when the dc jack is being used?