I have a small electronics repair shop out of my home and just yesterday checked in a third Toshiba machine with this same problem: failed GPU. Cause: overheating.
Each owner has said the same thing: over the past few months, every once in a while the screen stayed black during boot-up, but a restart or something similar like removing the battery, letting it sit overnight, or power-cycling made it boot correctly. Eventually nothing would help and they brought their laptops in for repair.
On the first machine I had to troubleshoot for quite a while before determining the issue, including everything mentioned here in this thread, until finally deciding to replace the mobo over re-soldering the GPU (which I incorrectly assumed was my only option). On the second unit I reflowed using a hairdryer, followed by my kitchen oven. Big job. Didn't want to have to do that with this machine, so I'm borrowing a heat gun tomorrow and using this procedure found on YouTube:
https://www.youtube.com/watch?v=cP2QOXNMA9g
(Warning: rated PG due to mild cursing)
This is a known issue with the laptops in this line and while one of the other suggestions might work and solve your specific problem, be on the watch for the graphics to eventually die due to dust and/or heat buildup in the future. It's HIGHLY recommended to use a cooling pad under these systems (some of the Toshiba User Manuals even go so far as to tell you that you NEED to use one rather than just SUGGEST it.) and to periodically tear-down and clean out the inside around the heatsink, fan, etc. You'll find the dustier or dirtier your normal environment, the more often you'll need to do it---at least twice per year is a good start, especially if you own pets that shed.
If you would like a copy of my t-shoot list to rule out all the other possible problems that can cause the black screen of doom, message me and I'll be glad to share it. Otherwise, assume your machine needs the gpu reflowed and don't trash it!
The hard disk, OS, screen, etc. are all still fine and in working order (unless you've inadvertently pressed the appropriate keys and don't something to the bios or something weird like that.)
IMO, Toshibas are good machines. I still have my original with Windows 95 on it! I own several models and have found, with the proper maintenance, that they'll last well past 10 years. (Believe it or not, I've found that Dells and Acers are the next best performers among abusive users. LOL.) Every manufacturer has weaknesses: none are perfect. Don't expect to not have issues with electronics because you'll always end up disappointed. Heat, dust and mishandling are the three biggest killers.
Good luck - I hope this helps some of you.
Mary
~Psalm 103~