Inequalities with determinants of perturbed positive matrices

We prove two inequalities regarding the ratio det(A+D)/detA of the determinant of a positive-definite matrix A and the determinant of its perturbation A+D. In the first problem, we study the perturbations that happen when positive matrices are added to diagonal blocks of the original matrix. In the second problem, the perturbations are added to the inverses of the matrices.