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If we define an operation that is $RCR_{ij}(A):=R_{ij}$, remove the $i^{th}$ column and the $j^{th}$ row of $A$, then this is expressible as  \begin{equation}  A^{-1}=\begin{bmatrix}   R_{11} |R_{11}|  & R_{12}J |R_{12}J|  & R_{13} |R_{13}|  \\ R_{21}J |R_{21}J|  & R_{22} |R_{22}|  & R_{23}J |R_{23}J|  \\ R_{31} |R_{31}|  & R_{32}J |R_{32}J|  & R_{33} |R_{33}|  \end{bmatrix}  \end{equation} 
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Clearly the condition for a right multiplication of the $J$ matrix being $i+j=odd$.  Alternatively $A_{ij}^{-1}=R_{ij}J^{i+j-1}$ $A_{ij}^{-1}=|R_{ij}J^{(i+j-1)}|$