I have recently come across some examples of matrices with a special structure. I will describe these matrices here and I hope that somebody will be able to point out a source where I can find more information about them. Consider an $n\times n$ matrix $A$ with elements $a_{ij}$ having the following properties. The elements with $i=j$ (call them $b_i$) are negative. The elements with $j=i+1 {\rm mod} n$ (call then $c_i$) are positive. All other elements are zero. The determinant of a matrix of this type is $\sum_i b_i+(1)^{n+1}\sum_i c_i$. A property of these matrices which I found surprising is that $(1)^{n+1}(\det A)A^{1}$ is a positive matrix, i.e. all its entries are positive. I found this by playing around with some examples. Can anybody point out to me some general theory which explains this observation? I met these matrices repeatedly when looking at certain chemical reaction networks. In that context the positivity statement is valuable because it allows the PerronFrobenius theorem to be applied.

A class of matrices with entrywise positive inverses (inversepositive matrices) appears in a variety of applications and has been studied by many authors. See, for example, http://ac.elscdn.com/002437959300244T/1s2.0002437959300244Tmain.pdf?_tid=9e7b73aff99f6a5e36bde4b1265a396c&acdnat=1332062723_3c39ada8e0b1d67ca7f0aedf2034ae45 or http://www.math.temple.edu/~abed/JS07.pdf 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 $(1)^{n+1} (1)^{ij} \det A_{ij} > 0$. 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 $i1$ and $ni$. The first block is upper diagonal with diagonal entries $b_1 b_2 \cdots b_{i1}$; the second block is lower diagonal with diagonal entries $c_{i+1} c_{i+2} \cdots c_n$. So $\det A_{ij}$ has sign $(1)^{i1} = (1)^{n+1} (1)^{ni}$ as desired. In particular, we have an explicit formula for $A^{1}$. The entry $(A^{1})_{ij}$ is $$ (1)^{ij} \frac{b_{j+1} b_{j+2} \cdots b_{i1} c_{i+1} c_{i+2} \cdots c_j}{\det A}.$$ 


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

