Inherited Matrix Entries: Principal Submatrices of the Inverse

For a nonsingular n-by-n matrix A = [aᵢⱼ ], let α ⊆ \ 1,2, ⋯ ,n\ and let A[α ] denote the principal submatrix of A lying in the rows and columns indicated by α. We determine the combinatorial circumstances under which the $(i,j)$ entry of the Schur complement (A- 1 [α ])- 1 equals aᵢⱼ, and under which the graph of this Schur complement is contained in the graph of A[α ].