You should try the Foundation as you will read on the website the genealogy office before putting their records with the Foundation on line will have accessed all local sources, parish registers etc. Ireland is a difficult place to search as most records were destroyed in 1922 in a Fire in Dublin. Its worth $10. CA to find out.
PRONI Public Records Northern Ireland has I understand parish records but I think personal research only.www.familysearch.org
has some records from early on. Go to website - don't use the first page but scroll down to UK/Ireland, click and go to next page and Ireland (7) I've never had much luck with this but worth a try.
I spent 4 visits to Ireland and can only get back to 1811.
So it isn't unusual to had a dead end there. You can be almost guaranteed if there first on has a name different from Matthew's then his father would be that name. In my husbands family they had a William as oldest son for generations right up to about 4 years ago.
If you ever locate a place you could look for Estate records
as I was able to trace back another 50 years for some family members.
Good luck. Annette