If we then have some matrix of individual particle states corresponding to \(\phi_{nm}=\phi_m(r_n)\) \[A=\begin{bmatrix} \phi_{11} & \phi_{12} & \phi_{13} \\ \phi_{21} & \phi_{22} & \phi_{23} \\ \phi_{31} & \phi_{32} & \phi_{33} \end{bmatrix} \\ \\ |A|=\Psi(r_1,r_2,r_3)=\frac{1}{\sqrt{6}}||\phi_1\phi_2\phi_3||\] we can calculate the inverse as \[A^{-1}=\frac{1}{|A|} \begin{bmatrix} \begin{vmatrix} \phi_{22} & \phi_{23} \\ \phi_{32} & \phi_{33}\end{vmatrix} & \begin{vmatrix} \phi_{13} & \phi_{12} \\ \phi_{33} & \phi_{32}\end{vmatrix} & \begin{vmatrix} \phi_{12} & \phi_{13} \\ \phi_{22} & \phi_{23}\end{vmatrix} \\ \begin{vmatrix} \phi_{23} & \phi_{21} \\ \phi_{33} & \phi_{31}\end{vmatrix} & \begin{vmatrix} \phi_{11} & \phi_{13} \\ \phi_{31} & \phi_{33}\end{vmatrix} & \begin{vmatrix} \phi_{13} & \phi_{11} \\ \phi_{23} & \phi_{21}\end{vmatrix} \\ \begin{vmatrix} \phi_{21} & \phi_{22} \\ \phi_{31} & \phi_{32}\end{vmatrix} & \begin{vmatrix} \phi_{12} & \phi_{11} \\ \phi_{32} & \phi_{31}\end{vmatrix} & \begin{vmatrix} \phi_{11} & \phi_{12} \\ \phi_{21} & \phi_{22}\end{vmatrix} \end{bmatrix} = \begin{bmatrix} \Psi(r_2,r_3) & \begin{vmatrix} \phi_{13} & \phi_{12} \\ \phi_{33} & \phi_{32}\end{vmatrix} & \begin{vmatrix} \phi_{12} & \phi_{13} \\ \phi_{22} & \phi_{23}\end{vmatrix} \\ \begin{vmatrix} \phi_{23} & \phi_{21} \\ \phi_{33} & \phi_{31}\end{vmatrix} & \Psi(r_1,r_3) & \begin{vmatrix} \phi_{13} & \phi_{11} \\ \phi_{23} & \phi_{21}\end{vmatrix} \\ \begin{vmatrix} \phi_{21} & \phi_{22} \\ \phi_{31} & \phi_{32}\end{vmatrix} & \begin{vmatrix} \phi_{12} & \phi_{11} \\ \phi_{32} & \phi_{31}\end{vmatrix} & \Psi(r_1,r_2) \end{bmatrix}\]
We also then know that \[\begin{bmatrix} \phi_{11} & \phi_{12} & \phi_{13} \\ \phi_{21} & \phi_{22} & \phi_{23} \\ \phi_{31} & \phi_{32} & \phi_{33} \end{bmatrix} \begin{bmatrix} \Psi(r_2,r_3) & \begin{vmatrix} \phi_{13} & \phi_{12} \\ \phi_{33} & \phi_{32}\end{vmatrix} & \begin{vmatrix} \phi_{12} & \phi_{13} \\ \phi_{22} & \phi_{23}\end{vmatrix} \\ \begin{vmatrix} \phi_{23} & \phi_{21} \\ \phi_{33} & \phi_{31}\end{vmatrix} & \Psi(r_1,r_3) & \begin{vmatrix} \phi_{13} & \phi_{11} \\ \phi_{23} & \phi_{21}\end{vmatrix} \\ \begin{vmatrix} \phi_{21} & \phi_{22} \\ \phi_{31} & \phi_{32}\end{vmatrix} & \begin{vmatrix} \phi_{12} & \phi_{11} \\ \phi_{32} & \phi_{31}\end{vmatrix} & \Psi(r_1,r_2) \end{bmatrix} = \Psi(r_1,r_2,r_3) \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\]