Why is the output voltage inverted when its amplified?
First, the voltage is only inverted by a common cathode gain stage where the the output voltage is taken off the plate. You also need to know that AC voltage and AC current are 180° out of phase. The reason the voltage is inverted is because the voltage on the grid controls the current through the tube. Since the input voltage and output current are in phase with each other, the output voltage with therefore be 180° out of phase with the input voltage.