Let $B$ denote the $n\times n$ matrix $I_x-A$. Let $[n]=\{1,2,\ldots,n\}$. For $I,J\subset [n]$ with the same number of elements, let $D_{I,J}$ denote the determinant of the matrix resulting from $B$ by deleting the rows indexed by the elements of $I$ and the columns indexed by the elements of $J$. Assuming $i\neq j$, we clearly have $P=D_{\varnothing,\varnothing}$, $P_i=D_{\{i\},\{i\}}$, $P_j=D_{\{j\},\{j\}}$, and $P_{ij}=D_{\{i,j\},\{i,j\}}$. By the <a href="https://en.wikipedia.org/wiki/Dodgson_condensation">Dodgson condensation identity</a>, $$ D_{\{i\},\{i\}}D_{\{j\},\{j\}}-D_{\{i\},\{j\}}D_{\{j\},\{i\}}= D_{\{i,j\},\{i,j\}}D_{\varnothing,\varnothing}\ . $$ Since $B$ is symmetric $D_{\{i\},\{j\}}=D_{\{j\},\{i\}}$. As a result $$ \Delta_{ij}=(D_{\{i\},\{j\}})^2\ . $$