Wow that's an old card, it's also a needlessly complex one.

Make sure that all 3 of it's modules are sat in their sockets correctly, could be something as simple as one of them coming loose.

Sorry there has been a mistake :/
The graphics card is ATi Radeon HD 5450 Video Card
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http://i.imgur.com/mKz5YXQ.jpg

So the real question is, what does "The computer won't work at all" mean?

It won't power? It won't post? It won't boot? No image to screen? What happens when you press power button? Any LED flash, sound made, do fans spin up?

It's doesn't post, boot, or power at all.
It doesn't response at all And no LED or fan spin etc

PSU failure.

I have already replaced my Power supply, but still not working

Then it grilled your Mobo and all your other components. Get a new Computer.

You can't just assume thing Mongius.

Go ahead and unplug everything, then plug it all back in. There should be a lit LED on your motherboard, and if there isn't one, then yes, your motherboard is fried. If it has lit up however, I would probably suggest replacing your CMOS battery, then go from there.

It's VERY unlikely to be a mobo or BIOS problem if it works perfectly fine without the GPU plugged in (a problem like that would usually be persistent). If you have spare parts (or you have a mate who does) try the GPU in another mobo, and a different GPU in your mobo.

That should tell you whether the GPU is dead (more likely) or the PCI controller is dead (less likely).

You might also want to think if there's any other factor that is changing between your mobo and GPU outputs (for example maybe your mobo uses VGA and your GPU uses DVI-D, in which case maybe it's a cable or monitor problem).

Although if it was a monitor failure it would power on right?