The study of food chains is an important topic in the determination of the spread and accumulation of environmental pollutants in living matter. Suppose that a food chain has three links. The first link consists of vegetation of types v₁, v₂. . . v_n, which provide all the food requirements for herbivores of species h₁, h₂, . . . , h_m in the second link. The third link consists of carnivorous animals c₁, c₂. . . ck , which depend entirely on the herbivores in the second link for their food supply. The coordinate ai j of the matrix

Represents the total number of plants of type vi eaten by the herbivores in the species h_j , whereas b_{i j} in

Describes the number of herbivores in species hi that are devoured by the animals of type c_j .

a. Show that the number of plants of type vi that eventually end up in the animals of species c_j is given by the entry in the ith row and jth column of the matrix AB.

b. What physical significance is associated with the matrices A⁻¹,B⁻¹, and (AB) ⁻¹ = B⁻¹A⁻¹?

an a12a bix .bn bva bil bn 2 b1 b2 bk bm b2bmk