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

Let B ∈Mn(C) be a row diagonally dominant matrix, i.e., σi|bii| = n ∑ j=1 j 6=i |bij |, i = 1, . . . , n, where 0 ≤ σi < 1, i = 1, . . . , n, with σ = max1≤i≤n σi. 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.