  Recently, Jel\'{i}nek derived that the number of self-dual interval orders of
reduced size $n$ is twice the number of row-Fishburn matrices of size $n$ by
using generating functions. In this paper, we present a bijective proof of this
relation by establishing a bijection between two variations of upper-triangular
matrices of nonnegative integers. Using the bijection, we provide a
combinatorial proof of the refined relations between self-dual Fishburn
matrices and row-Fishburn matrices in answer to a problem proposed by
Jel\'{i}nek.
