Hello! Grodno was a district (capital) city of Grodnenskaya gubernia of Russian Empire.
It's a National historical archive of Belarus in Grodno.
To take info from there you need to provide his name, surname and prob. father's name appr date of birth, religion and place of birth.
As i understand - his father named Matthew - it's non-slavic name.
You need to:
1) find his slavic name. I think it maybe Matvei (Matvej, Matvey, Matfei) - Матвей.
(full name Матвей Дмитрук)
2) find immigration records of his father on Ellis Island databese or Ancestry.com to find his appr. date of birth.
It maybe this one: http://search.ancestry.com/cgi-bin/sse.dll?gl=&gst=&...
(Find Matthew Dmitruk)
3) send a request to NIAB in Grodno. They will ask you for pre-paid and trying to search.