  We present analytic expressions for three-flavor neutrino oscillations in
presence of invisible neutrino decay and matter effects. Using the well-known
Cayley-Hamilton formalism, the leading-order terms are derived for oscillation
probabilities in all major channels assuming the neutrino mass eigenstate
$\nu_3$ to decay into sterile neutrinos. Our work extends and complements
previous studies utilizing the Cayley-Hamilton theorem, providing the series
expansions for $\nu_e \rightarrow \nu_e$, $\nu_e \rightarrow \nu_\mu$, $\nu_e
\rightarrow \nu_\tau$, $\nu_\mu \rightarrow \nu_\mu$, $\nu_\mu \rightarrow
\nu_\tau$ and $\nu_\tau \rightarrow \nu_\tau$. The accuracy of the analytical
formulas is investigated by comparing the results with numerically calculated
probabilities. We also comment on the implications on unitarity violation.
