key starts sometimes on 1990 Chevrolet Cavalier

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The key guy said that the 1990 Chev Cavaliers key assembly was not electrical. Long ago he put graphite and that seemed to work. But recently, not even graphite worked. He said it was not the key mechanism and suggested it might be the battery. I had the battery checked and the battery was good. The parts place guy said that it may be the starter. Does somebody know the correct answer? If you let the car set for a while...such as will start. But if you drive a while then turn the engine may or may not start. The key turns and you can hear other electrical stuff, fans turn on....but the car won't crank.
(1) Answer
Your starter is not working when the engine is hot. But it works when the engine is cold. So, see if you are getting a 12 volt signal from the Ignition Switch to the purple wire on the starter solenoid when you turn the key when the car won't start. If you are, then it is most likely the starter. When GM starters go, they often will not crank when the car is hot.