Is it possible to add a related matrix field and show the first entry in that field on the http://domain.com/admin/entries section of the control panel?
That's not possible, Matrix blocks (and also Tags) don't have an element index view in the CP.
You could write a plugin that adds such a page to the CP. It would list all the Matrix blocks that meet your requirements (owner ID, owner entry, ...).
Another idea would be to just use an Entries field instead; the entries section you'd set up for the field comes with an index view for free.