You don't really define what you mean by a row being greater than another (for instance, what if the first column is larger and the second column is smaller?). However, if you wanted to determine if each successive element in a column of your data is smaller than the next, you could use the diff
function:
dfSource$larger <- c(diff(dfSource[,1]) > 0, FALSE)
dfSource
# X1 X2 X3 larger
# row1 1 2 3 FALSE
# row2 1 2 3 TRUE
# row3 2 3 4 FALSE
# row4 2 3 4 FALSE