Matilda's family in 1861 
RG 9; Piece: 2184; Folio: 14; Page: 22
Edward Plant   46  Frying Pan Maker    Birmingham, Warwickshire
Maria Plant     47                              Birmingham, Warwickshire
Maria Plant          15                              Birmingham, Warwickshire 
Resehannah Plant   12                     Birmingham, Warwickshire 
Priscilla Plant   7                               Birmingham, Warwickshire
Matilda Plant   5                               Birmingham, Warwickshire
I'm afraid you are not likely to find why they left - as I said it was probably just for a better life.