Sum of Elements of Invertible Matrix

Theorem
Let $\mathbf J_n$ be the $n \times n$ square ones matrix.

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

Then:


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

Lemma
Let$\mathbf J_n$ be the $n \times n$ square ones matrix.

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

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

Then:

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

Then:

Then: