This is a quick guide to hacking the PlayStation 3 with the TI-84. Most of the guides I've read spend way too much time talking about irrelevant stuff and going into long-winded technical explanations that you probably don't want to bother about. I have yet to see a guide that simply gives step-by-step instructions on how to hack the PS3, which is why I'm creating this.
Note: I am only posting this on an educational basis. If you mess up and it breaks your PS3, tough luck. I cannot be held responsible for anything that happens to your PS3.
PS3 hacking will only work if you have firmware 3.41 or below. Anything higher, and you won't be able to hack your PS3. You also cannot downgrade from firmware 3.42 to 3.41. This is because the firmware is flashed onto the PS3 itself and not the harddrive (this also explains why the PS3 firmware stays the same when you replace the harddrive). Whatever you may have read on another blog of forum, let me reiterate that you cannot downgrade from firmware 3.42 on the PS3.
My PS3 is currently on firmware 3.41, and the method I'm outlining worked without any issues. This PS3 jailbreak should also work with firmware 3.40, but I'm not sure about the rest. I do not personally know much about the differences in firmware in respect to the jailbreak, but I guess you will find all the information you need if you search around a bit on Google and on any PS3 forum (which won't take long because that's all they're talking about these days).
If you have anything lower, I advise you download the 3.41 firmware (you know the drill) and install it on your PS3.
Once you've downloaded the firmware, place it in X:\PS3\UPDATE on a USB drive to update your PS3 (do a manual backup first). If you usually update your firmware via USB, you're probably used to this. If you usually update directly on your PS3, you're doing it wrong. It is better and safer to update your PS3 via USB. Not that this matters now because you obviously won't be able to update your PS3 again once it's jailbroken.
You can use various devices to hack your PS3. This device compability list at PSFreedom is the best such list. Do not trust anything else. If it's not there, it cannot be done. Obviously, if you do not have any of those devices, you won't be able to hack your PS3. These instructions work with the TI-84 only.
To jailbreak your PS3, download the app from PS3JB and follow the instructions. PS3JB has the best instruction for hacking the PS3 with a TI-84 and has all the screenshots you need.
Once you've hacked your PS3, you will also need to download the PS3-Hacks Backup Manager. Place the Backup Manager on a USB drive, insert it in the PS3. From here, choose Install Package Files, and install the Backup Manager. You're done!
Here's a video of someone using this method to successfully hack a PS3. It's a fairly quick process, and you'll be able to back up your games on your PS3 once you're done. However, remember that you will not be able to use PSN or update your firmware once your PS3 is hacked.