I need help with this, i think you are supposed to use matrices or something

Question

Assume this is a closed Leontief model. We assume that in a village there are farmers, carpenters, and tailors, who provide the three essential goods: food, housing, and clothing. Suppose the farmers consume 2/5 of the food (produced by farmers), 1/3 of the housing (produced by carpenters) and 1/2 of the clothes (produced by tailors). The carpenter consumes 2/5 of the food, 1/3 of the housing, and ⅕ of the clothes. The tailors consume ⅕ of the food, 1/3 of the housing, and no clothes.

If we know that the tailors produce 72 units of clothes, then the farmers produce ___ units of food and the carpenters produce ___ units of housing.

Answer 1

For a closed model, matrix notation,

AX=X or AX=IX

For this case,

| 2/5 1/3 1/2 |

A = | 1/5 1/3 1/5 |

| 1/5 1/3 0 |

| 3/5 -1/3 -1/2 |

I-A = | -1/5 2/3 -1/5 |

| -1/4 -1/3 1 |

Using Gauss-Jordan elimination

x=3z/2 = 108

y=6z/5 = 86.4

z=72 which was given