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In this paper we derive new properties complementary to an $2n\times 2n$ Brualdi-Li tournament matrix $B_{2n}$. We show that $B_{2n}$ has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of $B_{2n}$ is also determined. Related results obtained in previous articles are proven to be corollaries.