Matrices whose inverse is positive

A class of matrices with entrywise positive inverses (inverse-positive matrices) appears in a variety of applications and has been studied by many authors. See, for example,

M-Matrices Whose Inverses Are Totally Positive

or

Positive, path product, and inverse M-matrices

In these papers (and those referred to by them) you will find methods to construct other classes of matrices with this property.

This is straightforward from the adjoint formula for the inverse matrix. Let $A_{ij}$ be the matrix formed by deleting row $i$ and column $j$ from $A$. We must show that $(-1)^{n+1} (-1)^{i-j} \det A_{ij} > 0$.

We can reorder the rows and columns of $A$ cyclically to assume without loss of generality that $j=n$. Then $A_{in}$ is block diagonal with two blocks of size $i-1$ and $n-i$. The first block is upper diagonal with diagonal entries $b_1 b_2 \cdots b_{i-1}$; the second block is lower diagonal with diagonal entries $c_{i+1} c_{i+2} \cdots c_n$. So $\det A_{ij}$ has sign $(-1)^{i-1} = (-1)^{n+1} (-1)^{n-i}$ as desired.

In particular, we have an explicit formula for $A^{-1}$. The entry $(A^{-1})_{ij}$ is $$ (-1)^{i-j} \frac{b_{j+1} b_{j+2} \cdots b_{i-1} c_{i+1} c_{i+2} \cdots c_j}{\det A}.$$

Matrices whose inverses are nonnegative are also called monotone. There are a number of equivalent characterizations in Theorem 6.2.3 of the wonderful book by Berman and Plemmons:

http://books.google.ie/books/about/Nonnegative_Matrices_in_the_Mathematical.html?id=MRB7SUc_u6YC&redir_esc=y

In the case of this question, the matrix might not be an $M$-matrix. It depends on the actual entries.