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Can a Graphics card even instantly make a computer turn off instantly?
Yes
(not even BSOD?)
Bluescreen is usually some sort of memory fault.
Do you think the PSU is broken and can't power the Graphics Card?
Unlikely, worth a try anyway. Have another GPU that you know works to test it?
Or is it the Graphics Card itself?
Way more likely.
According to byt411, if I wait a few days, it'll work again. Is that possible?
No. Electrical systems dont just heal themselves over time. Either you blew a fuse and it will reset after ~1 hour or you killed your GPU. Another possibility is that you bricked your GPU. Its electricity is fine, but the software is fucked up.
If you can turn on your computer properly, then there isn't a problem.
That happened to me once; MSI Afterburner wouldn't detect it or anything, so all I did was I ignored it, and a few days later, it worked again.
He cant turn his machine on properly. Fallback to onboard GPU is not "turn on properly".
Id take the GPU out and test it on another machine if possible. While you have it out see if you find any burnmarks or other defect parts.
In case its not obvious: the last one is a joke. If your GPU looked like this youd know allready.