Sum of Elements of Invertible Matrix

Theorem
Let $J_n$ be the $n\times n$ matrix of all ones.

Let $B$ be an $n\times n$ invertible matrix with entries $b_{i j}$, $1 \le i, j \le n$.

Then:


 * $\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n b_{i j} = 1 - \map \det B \, \map \det {B^{-1} - J_n}$

Lemma
Let $J_n$ be the $n\times n$ matrix of all ones.

Let $A$ be an $n \times n$ matrix.

Let $A_{ij}$ denote the cofactor of element $a_{ij}$ in $\map \det A$, $1 \le i, j \le n$.

Then:

Let $A = B^{-1}$.

Then:

Then: