Bounds on determinants of perturbed diagonal matrices

We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If A = I−E is a real n×n matrix and the elements of E are bounded in absolute value by ε ≤ 1/n, then a lower bound of Ostrowski (1938) is det(A) ≥ 1−nε. We show that if, in addition, the diagonal elements of E are zero, then a best-possible lower bound is det(A) ≥ (1− (n− 1)ε) (1 + ε).