My s3 mini won't turn on

A

AC Question

My s3 mini has been getting slow of late and I've had to repeatedly clear RAM to fix the issue. Also, the battery life barely lasts a day. Whilst doing homework, I was listening to music on the youtube app on my phone when it froze. I carried on doing the homework for a few more minutes before remembering that it wasn't playing and it hadn't sorted itself out. I pressed the power button a few times to go to sleep and back to the homepage again but it didn't do anything for a few seconds. Then it turned black and waited the usual length of time before vibrating to let me know it had shut down. Since then, I have not been able to:

1) Turn it on (the power button has a "good clicky sound" but no lights, no vibration, no screen turning on)
2) Charge it up (It had over 15% battery left when it turned off as the screen was on full brightness but the charging cable yields no animation on the screen or vibration
3) Connect it to a computer (I know that this is one wasn't going to happen as the phone isn't on but it was worth a try - the computer doesn't recognise that there is anything connected)
4) Enter any modes (safe mode, download mode or any other mode I've tried on the internet cannot be reached as the power button is effectively doing nothing)

I have tried removing the battery and holding down the power button to drain any residual charge from the capacitors then replace the battery but no joy. Not a single light turns on. I swapped batteries with my friend who has an s3 mini and his turned on but mine didn't. I also tried my old galaxy y for the computer connection which was recognised where my s3 mini wasn't.

I'm starting to believe that it must be a motherboard problem because although the screen isn't that damaged, the rest has been damaged quite a bit but would welcome any opinions either way.