Arduino + Max7219 only works if...

Rset is a bit high in my opinion. Choose a 62 K and see what happen.

In the void setup(), the MAX7219 is not properly setup in my opinion. It is missing a few init line. Check the datasheet of the MAX7219 if you have question.

If your setup was working connect to the PC, and connect to the 9V battery is not, too much current is being use. ( Over 100 mA ). To confirm it, measured the current going into the 2.1 mm jack.