We continue the study of matrices over a supertropical algebra, proving the
existence of a tangible adjoint of $A$, which provides the unique right (resp.
left) quasi-inverse maximal with respect to the right (resp. left)
quasi-identity matrix corresponding to $A$; this provides a unique maximal
(tangible) solution to supertropical vector equations, via a version of
Cramer's rule. We also describe various properties of this tangible adjoint,
and use it to compute supertropical eigenvectors, thereby producing an example
in which an $n\times n$ matrix has $n$ distinct supertropical eigenvalues but
their supertropical eigenvectors are tropically dependent.