I drove my 2000 mitsubishi mirage with the a/c knowing the compressor is broken.
Air conditioner compressors are not engaged unless the air conditioning is switched ON. Mounted to the front of the A/C compressor is an electro-magnetic clutch. Meaning, the clutch and belt spin freely without the internals of the compressor engaged and being moved along with the clutch pulley. Once you switch the A/C ON, then the clutch is energized electrically to engage the internals of the compressor and spins them along with the clutch pulley.
As long as the A/C compressor is not physically damaged (broken or seized), it should not effect the normal operation of the engine and transmission. If, however, when you switch the A/C ON, and the car begins to run poorly, then I suggest you do not operate the air conditioning until you have the AC system diagnosed and AC compressor replaced if necessary.
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