When the GPU is dead, normally the machine does not make it to the login screen.
I'd suggest to do the following:
- insert a blank USB thumb drive, ~ 16GB
- go into "Internet recovery", and re-install macOS onto the thumb drive. That way, you don't need to re-install your existing machine, and you can get to the original state by simply unplugging the thumb drive.
- boot from the thumb drive.
If this works, it is simply a software issue. If this also not works, it will be something from the hardware.
You could also run Apple Diagnostic Software (Apple ASD145A for 2011 iMac). I find this more complex than installing onto a thumb drive.
The key combinations for internet recovery and 'boot from other disk' can be found in the web.