Matrix commutation with an inverse ($AB=BA⟹A{B}^{-1}=B^{-1}A$)

If two square matrices satisfy $AB=BA$ and $B$ is invertible, multiplication on the left and right by $B^{-1}$ gives $B^{-1}A=A{B}^{-1}$. Consequently a polynomial in a matrix commutes with the matrix inverse of any invertible polynomial in the same matrix.