It is unfortunate that the HDMI is not getting detected. This could get fixed by updating drivers. You may use the below link to update Intel and NVIDIA drivers:
http://dell.to/13SOQR7 NVIDIA Windows 7 and Windows 8
If the computer has any other Operating system apart from Windows 7 then you may use the below link:
Also make sure changes are made in TV input source so that it gets detected in system control panel.
If you have further questions do reply to the same post.
Thank you so much for updating me on this. I request you to use the below link to check if the computer is still in warranty:
Also update me if the technician advised replacement of motherboard.
If the computer is in warranty please send me the system service tag through private message so that I can check and let you know what needs to be done.
To send a private message, click on my name and select start conversation.
I would still recommend you to contact the out of warranty team on 18002884410 (U.S only). They can help you further. If you belong to a different region do let me know.