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.



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](https://core.ac.uk/download/pdf/82554224.pdf)


or


[Positive, path product, and inverse M-matrices](https://core.ac.uk/download/pdf/81962688.pdf)


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


