Supertropical Matrix Algebra Ii: Solving Tropical Equations Zur Izhakian and Louis Rowen

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 × n matrix has n distinct supertropical eigenvalues but their supertropical eigenvectors are tropically dependent.