On the zero set of the Kobayashi--Royden pseudometric of the spectral unit ball

Given $A\in {\mathrm{\Omega }}_n,$ the $n^2$-dimensional spectral unit ball, we show that $B$ is a “generalized” tangent vector at $A$ to an entire curve in ${\mathrm{\Omega }}_n$ if and only if $B$ is in the tangent cone $C_A$ to the isospectral variety at $A\mathrm{.}$ In the case of ${\mathrm{\Omega }}_3,$ the zero set of this metric is completely described.