Relation between the sum of principal minors of different orders

Question

Let $A$ be a symmetric (0,1)-square matrix of order $n$ having the diagonal entries zero. Let $m$ be the nullity of $A$ (number of zero eigenvalues), denoted by $\eta(A)$. Let $A_1$ be the square matrix of order $n-1$, resulting after deleting the last row and the last column of $A$. Assume that $\eta(A_1)=\eta(A)+1$. Let $S_1$ be the sum of all the principal minors of order $n-\eta(A)$ in $A$ and $S_2$ be the sum of all the principal minors of order $n-\eta(A)-2$ in $A_1$. Then prove that $S_1\neq -S_2$.