On the characteristic polynomial of a supertropical adjoint matrix

Let $\chi (A)$ denote the characteristic polynomial of a matrix $A$ over a field; a standard result of linear algebra states that $\chi (A^{-1})$ is the reciprocal polynomial of $\chi (A)$. More formally, the condition $\chi^n(X) \chi^k(X^{-1})=\chi^{n-k}(X)$holdsforanyinvertible$n\times n$matrix$X$overafield,where$\chi^i(X)$denotesthecoefficientof$\lambda^{n-i}$ in the characteristic polynomial $\det (\lambda I-X)$. We confirm a recent conjecture of Niv by proving the tropical analogue of this result.