# Real world definition of the inverse of a matrix

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.

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Could you at least please try to use LaTeX for the math? You've been here for almost a year now and this was terribly written. I also took the liberty to fix up your grammar. –  kahen Oct 15 '11 at 13:23
Even after kahen's cosmetics, I'm still not sure what to make of this question; none of the matrices under consideration seem to be square... –  Guess who it is. Oct 15 '11 at 13:26
@Kahen i dont know how to format matrix forms –  dato datuashvili Oct 15 '11 at 13:30
You might want to use this to help you with making $\TeX$-ed up matrices... –  Guess who it is. Oct 15 '11 at 13:32
i will use it,sorry –  dato datuashvili Oct 15 '11 at 13:42

For example, this is what I understand from the relation you are imposing only because the size of the involved variables match: $$v=Ah \quad ,\quad h=Bc \implies v=ABc$$ where $c,v,h$ are the column vectors consisting $v_i,c_i$ and $h_i$ stacked on top of each other.
But what are the quantities? what is the dimension of say entries of $A$, $\frac{plant}{animal}$? If I multiply $Ah$ the result is the total number of plants eaten by the herbivors and that is not the variable $v$ that you defined in the beginning.