  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 $\chi^n(X)
\chi^k(X^{-1})=\chi^{n-k}(X)$ holds for any invertible $n\times n$ matrix $X$
over a field, where $\chi^i(X)$ denotes the coefficient of $\lambda^{n-i}$ in
the characteristic polynomial $\det(\lambda I-X)$. We confirm a recent
conjecture of Niv by proving the tropical analogue of this result.
