Estimates of the determinant of a perturbed identity matrix

Recently Brent et al. presented new estimates for the determinant of a real perturbation I+E of the identity matrix. They give a lower and an upper bound depending on the maximum absolute value of the diagonal and the off-diagonal elements of E, and show that either bound is sharp. Their bounds will always include 1, and the difference of the bounds is at least tr(E). In this note we present a lower and an upper bound depending on the trace and Frobenius norm ϵ:=‖E‖F of the (real or complex) perturbation E, where the difference of the bounds is not larger than ϵ2+O(ϵ3) provided that ϵ<1. Moreover, we prove a bound on the relative error between det⁡(I+E) and exp⁡(tr(E)) of order ϵ2.