Supertropical Matrix Algebra II: Solving tropical equations

Understand

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.