Alienware X51 R1 not recognizing graphic card

If I may ask, if you are using an HDMI cable, where do you have it hooked up?

I'm not using an HDMI cable.  I use a VGA cable plugged into the video card port on the back of the computer.

That makes things worse.

Are you able to install the AMD drivers again? The AMD GPU could be a hidden device.
Have you also tried reseating the GPU?

Why does it make things worse?

I've hooked my PC up to my TV using an HDMI cable and I get the same problem. When I swap out my i7-3770 for the i3-2120 it replaced, the video card works like normal.  I'm not able to update drivers since the PC doesn't seem to recognize the GPU at all. When I try to do an auto-detect on the AMD site it doesn't detect the card either. The problem has to be related to the new CPU in some way since that is the only thing I've changed.

My card is an AMD Radeon 7700.

I viewed as worse because I thought this could be a bit more simple.

However, since you went from Sandy-Bridge to Ivy-Bridge, I assume you updated the BIOS? Does the BIOS even detect the GPU?

I'm using the a06 version of the bios. How do I tell if my bios is detecting the gpu?

There should be a row that says discrete grpahics card once you enter into the BIOS. It should say AMD Radeon GFX or something like that (for Nvidia GPUs, it's Nvidia GFX).

I would update to a more recent BIOS to see if that does the trick.

Run a full system check Enhanced Systems Diagnostics (ePSA)

ePSA can be selected from the BIOS Boot Menu under Diagnostics, in other words:

1. Press at the welcome screen for the boot menu

2. At the One Time Boot menu, press the arrow key to highlight Diagnostics,
    then press to begin ePSA.

Alienware Systems Diagnostics Link ☛ 

I've updated to A07 and still have the same problem.  I have an 08pg26 motherboard.  It looks like the BIOS isn't even recognizing the video card.  Is there a list of video cards that are compatible with this motherboard and BIOS version?  Maybe I should just try replacing it.