A

Step 1: According to the ideal gas law, the relationship between pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas is given by the equation: $PV = nRT$.
Step 2: In this scenario, we have a cylinder filled with gas in a heat-proof jacket, implying that there is no heat exchange with the surroundings (adiabatic condition).
Step 3: When the volume of the cylinder is gradually increased while keeping the system adiabatic, the internal energy of the gas changes because the gas does work on the walls of the cylinder.
Step 4: As the volume increases, the pressure decreases since the amount of gas and the gas constant remain constant. Consequently, in an adiabatic process, the temperature of the gas decreases.
Step 5: Therefore, the temperature of the gas will decrease as the volume of the cylinder is gradually increased.