Auto login to LinkedIn I have installed HybridAuth module. I can login from my website using LinkedIn. But my question is, how can I login automatically into LinkedIn when I login from my website using LinkedIn login?

I have gone though various APIs but could not get proper information.
Got something closely related to your case and I won't say this is not possible, you can have some workaround by following these threads:

Login to Drupal using LinkedIn

This feature is not available right now, to login into Linkedin while using its credential on your site. But it seems that, this thread is in knowledge of Linkedin Officials as well and there may be some work is going on to provide you create app feature just like Facebook.

Here is something to elaborate the same: 

http://developer.linkedin.com/forum/how-create-linkedin-application-tripit-or-slideshare