On the characteristic polynomial of a supertropical adjoint matrix

Let χ(A)χ(A) denote the characteristic polynomial of a matrix A   over a field; a standard result of linear algebra states that χ(A−1)χ(A−1) is the reciprocal polynomial of χ(A)χ(A). More formally, the condition χn(A)χk(A−1)=χn−k(A)χn(A)χk(A−1)=χn−k(A) holds for any invertible n×nn×n matrix A   over a field, where χi(A)χi(A) denotes the coefficient of λn−iλn−i in the characteristic polynomial det⁡(λI−A)det⁡(λI−A). We confirm a recent conjecture of Niv by proving the tropical analogue of this result.