This paper presents several results associated to a holomorphic-interpolation problem for the spectral unit ball \(\Omega_n=\{W\in M_n({\mathbb C}): r(W)<1\}\), where \(n\geq 2\). It begins by showing that a known necessary condition for the existence of a \(\mathcal{O}(\mathbb{D};\Omega_n)\)-interpolant with (\(\mathcal{O}(\mathbb{D};\Omega_n)\) denoting the class of all holomorphic maps from the open unit disc \(\mathbb{D}\) centered at \(0\in\mathbb{C}\) into \(\Omega_n\)), given that the matricial data are non-derogatory, is not sufficient. The paper provides next a new necessary condition for the solvability of the two-point interpolation problem. The new necessary condition is not restricted only to non-derogatory data and incorporates the Jordan structure of the prescribed data. Finally, the paper proves a Schwarz-type lemma for holomorphic self-maps of \(\Omega_n\;(n\geq 2)\) by using some of the ideas used in deducing the latter result.