Let $A$ $\in$ $M_n (C)$. For each $1 \leq i \leq n$, let $A_i$ be the $(n-1)\times(n-1)$ principal submatrix of $A$ resulting from deleting the $i^\text{th}$ row and $i^\text{th}$ column. Prove that for $1 \leq k \leq n-1$, $\sum\limits_{i=1}^n E_{k}(A_{i}) = (n-k)E_{k}(A)$.
EDIT: The notation is consistent with Horn and Johnson's Matrix Analysis book. On page 40, it says that there are $\binom{n}{k}$ different k-by-k principal minors of the matrix $A=[a_{ij}]$, and the sum of these is denoted by $E_{k}(A)$
I have been trying different formulas involving the trace and the determinant, but I think I am missing some insight into this question. Any help will be appreciated, thanks.