The Jordan and Frobenius pairs of the inverse

Given a matrix $Ainmathbb{C}^{n imes n}$ there exists a nonsingular matrix $V$ such that $V^{-1}AV=J$, where $J$ is a very sparse matrix with a diagonal block structure, known as Jordan canonical form (JCF) of $A$. Assume that $A$ is nonsingular and that $V$ and $J$ are given. How to obtain $widehat{V}$ and $widehat{J}$ such that $widehat{V}^{-1}A^{-1}widehat{V}=widehat{J}$ and $widehat{J}$ is the JCF of $A^{-1}$? Curiously, the answer involves the Pascal matrix. For the Frobenius canonical form (FCF), where blocks are companion matrices, the analogous question has a very simple answer. Jordan blocks and companion are non-derogatory lower Hessenberg matrices. The answers to the two questions will be obtained by solving two linear matrix equations involving these matrices.