Ian's solution takes advantage of one of the properties of Header columns:It allows defining the 'whole row' using only the row number. The 'whole row' in this case does not include the cell in the header column, thus side-stepping the 'cell references itself' error that would rise if it were included.
Since your example shows the row's total being reported in the rightmost column, you will not be able to use that shortcut; you must use both column and row labels in the definitions of the first and last cells of the range to be summed.
Using your example, this is the formula you need in E2: =SUM(A2:C2)
Filled down to E7, the formula will automatically adjust to refer to the sme three columns in the row containing the formula.