Hi, as a furtheron from what Hume was saying, I believe that you are confused with information about the 2 brothers.
1861 census:
Grainery Row, Kirkintilloch, Dumbarton
George Hamilton age:27 born:1834 Stevenson, Ayr
occupation: cotton weaver
Elizabeth Hamilton wife age:25 born:1836 Kirkintilloch
Archibald Hamilton son age:5 born: Kirkintilloch (8 Oct 1856)
George Hamilton son age:8 months born: 1860 Kirkintilloch
1861 census:
Dandy Row, Old Monkland, Lanark
Matthew Hamilton age:18 born:1843 Stevenson, Ayr
occupation: collier
lodger with the McAlester family
1871 census:
Townland, Kirkintilloch, Dumbarton
Matthew Hamilton age:35 born:1836 Stevenson, Ayr
occupation: coal miner
Elis Hamilton wife age:30 born:1841 Glasgow, Lanark
Archibald Hamilton son age:11 born:1860 Kirkintilloch
1871 census:
George & Elizabeth Page Hamilton are still living in Kirkintilloch, Dumbarton with their family.
Immigration:
Matthew Hamilton age:35 born :1840
Elizabeth Hamilton age:33 born:1842
Archibald Hamilton age:15 born:1860
all left from Glasgow, Scotland
arriving in Brisbane, Australia
on 20 September 1875
on board the ship Renfrewshire
** found in the passanger lists for Queensland,Australia
Marriage:
8 May 1855 Kirkintilloch, Dumbarton
George Hamilton age:21 born:1834 1st April Stevenson, Ayr
occupation: weaver
father: Archibald Hamilton occ: tailor (dec.)
mother: Grace McKechan
Elisabeth Page age:19 born:1836 1st April Kirkintilloch, Dumbarton
occupation: domestic servant
father: James Page occ: tailor (dec.)
mother: Margaret Semple
Marriage:
29 November 1861
Matthew Hamilton age:23
occupation: collier
father: Archibald Hamilton occ: tailor (dec.)
mother: Grace McKechan (dec.)
Elizabeth Smith age:22
occupation: handloom weaver
father: John Smith occ: boatman (dec.)
mother: Mary Thornton
The confusion between George and Matthew is that both brothers married an Elizabeth- George to Elizabeth Page and Matthew to Elizabeth Smith. Matthew and his family did indeed migrate to Australia, maybe his brother George's son Archibald followed his uncle. I hope this helps clear up the confusion and gives you more info, Kim