2000 Monte Carlo SS, 3.8, sometimes when I try to start it it don't do anything. on 2000 Chevrolet Monte Carlo

The car runs great, but at random times I try to start it when I turn the key it does nothing. I don't click, turn over, nothing at all. When it happens there is power at the instrument cluster and the accessories work it just won't engage the starter. After turning it on and off constantly for ten minutes or so it'll start right up and be fine for a while, days, a week maybe but then it happens again. There is an issue with the fuel gauge at times, could be related. I'm guessing it's the switch itself but I need some advice on this issue.

When it doesn't start, test for B+ at the starter wire, usually purple, at the starter solenoid.
Thank you for the prompt reply, I really appreciate it. So, you mean run a jumper from the battery + to the solenoid and if it cranks over the switch is likely bad? Or are you saying that if there is juice at the solenoid it's likely the starter and/or the solenoid, or something else entirely but not the switch. If there is no power at the solenoid it would most likely be the switch, right?
Just checking to see if switch circuit is applying power to solenoid to verify if it's the starter or something on the switch circuit. Which direction
Makes sense, I'll try it the very next time it happens. Thanks again
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