Gaussian elimination is stable for the inverse of a diagonally dominant matrix

Let B E M n (C) be a row diagonally dominant matrix, i.e., σ i |b ii |=Σ|b ij |, i=1,...,n, where 0 ≤ σ i < 1, i = 1,...,n, with a = max 1≤i≤n σ i . We show that no pivoting is necessary when Gaussian elimination is applied to A = B -1 . Moreover, the growth factor for A does not exceed 1 + σ. The same results are true with row diagonal dominance being replaced by column diagonal dominance.