Let us consider an example from the real world: study of food chains, where there is very important determination of spread and accumulation of environmental pollutants in living matter. Suppose that the food chain has three links: The first link consist of vegetation of types $v_1,v_2,\dotsc,v_n$, which provides all the food requirements for herbivores of species $h_1, h_2, \dotsc, h_m$ in the second link. The third link consists of carnivorous animals $c_1, c_2, \dotsc, c_k$, which depend entirely upon the herbivores in the second link for their food supply.
Suppose a matrix
$A = [a_{ij}] = \begin{pmatrix} a_{11} & a_{12} & \dotsb & a_{1m} \\\\ a_{21} & a_{22} & \dotsb & a_{2m} \\\\ \vdots & \vdots & \ddots & \vdots \\\\ a_{n1} & a_{n2} & \dotsb & a_{nm} \end{pmatrix}$ represents the total number of plants of type $v_i$ eaten by the herbivores in the species $h_j$, and another matrix $ B = [b_{ij}] = \begin{pmatrix} b_{11} & b_{12} & \dotsb & b_{1k} \\\\ b_{21} & b_{22} & \dotsb & b_{2k} \\\\ \vdots & \vdots & \ddots & \vdots \\\\ b_{m1} & b_{m2} & \dotsb & b_{mk} \end{pmatrix}$ represents the number of herbivores in species $h_i$ which are devoured by the animals of type $c_j$.
My question is what does $A^{-1}$, $(AB)^{-1}$ and $B^{-1}$ represent?
Thanks a lot.