TITLE: Matrices whose inverse is positive
QUESTION [11 upvotes]: 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 
$\prod_i b_i+(-1)^{n+1}\prod_i c_i$. 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 Perron-Frobenius theorem to be applied.

REPLY [3 votes]: 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.