Inverse of a matrix related to double zeta values of odd weight

In this paper, we give a proof of a conjecture made by Zagier about the inverse of some matrix related to double zeta values of parity $(\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},\mathrm{o}\mathrm{d}\mathrm{d})$. As a result, we obtain a family of Bernoulli number identities. We further generalize this family to a more general setting involving binomial coefficients of negative arguments.