The national economy of Seriland manufactures three products: steel, cars, and machines. (...

The national economy of Seriland manufactures three products: steel, cars, and machines. (1) To produce $1 of steel requires 30¢ of steel, 15¢ of cars, and 40¢ of machines. (2) To produce $1 of cars requires 45¢ of steel, 20¢ of cars, and 10¢ of machines. (3) To produce $1 of machines requires 40¢ of steel, 10¢ of cars, and 45¢ of machines. During the coming year, Seriland wants to consume ds dollars of steel, dc dollars of cars, and dm dollars of machinery.

For the comine year, let

s = dollar value of steel produced

c = dollar value of cars produced

m = dollar value of machines produced

Define A to be the 3 × 3 matrix whose ijth element is the dollar value of product i required to produce $1 of product j (steel = product 1, cars = product 2, machinery = product 3).

a Determine A.

b Show that

(Hint: Observe that the value of next year’s steel production = (next year’s consumer steel demand) + (steel needed to make next year’s steel) + (steel needed to make next year’s cars) + (steel needed to make next year’s machines). This should give you the general idea.)

c Show that Equation (24) may be rewritten as

d Given values for ds, dc, and dm, describe how you can use (I − A)−1 to determine if Seriland can meet next year’s consumer demand.

e Suppose next year’s demand for steel increases by $1. This will increase the value of the steel, cars, and machines that must be produced next year. In terms of (I − A)−1 determine the change in next year’s production requirements.