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Now, let PF, PT, Pc, PCM, and PSB denote the incomes of the Farmer, Tailor, Carpenter, Coal Miner, and Slacker Bob, respectively. Note that each of these quantities not only denotes the incomes of each of our esteemed citizens but also the cost of the corresponding goods. So for example, PF is the Farmer's income as well as the cost of all the food. So if the Farmer produces $100 worth of food, then his income will also be $100 since all of this food is bought out and the profits go to the Farmer. The idea is, of course, to be able to figure out how we should price the goods in order for the citizens of Matrixville to survive; i.e., we must find PF, PT, Pc, PCM, and PSB subject to the following conditions: 0.25PF + 0.15PT + 0.25Pc - 0.18PCM + 0.20PSB = 0.15PF + 0.28PT - 0.18Pc - 0.17PCM - 0.05PSB = 0.22PF + 0.19PT - 0.22Pc + 0.22PCM - 0.10PSB = 0.20PF + 0.15PT + 0.20Pc + 0.28PCM - 0.10PSB = 0.18PF + 0.23PT - 0.18Pc - 0.17PCM - 0.50PSB = PSR Exercise 2.5: Explain where this system of equations came from and what it means. (Think about what the left-hand side and the right-hand side of each equation mean.) Let us denote the column vector (PF, PT, Pc, PCM, PSB) by P, and let E be the coefficient matrix of the above system. We can now rewrite that system as: EP = P equivalently EP - P = (E - I)P = 0 where I is the 5x5 matrix with 1's on the diagonal and 0's everywhere else. The property of the matrix we are using is that Iv = 0 for any vector.

          Now, let PF, PT, Pc, PCM, and PSB denote the incomes of the Farmer, Tailor, Carpenter, Coal Miner, and Slacker Bob, respectively. Note that each of these quantities not only denotes the incomes of each of our esteemed citizens but also the cost of the corresponding goods. So for example, PF is the Farmer's income as well as the cost of all the food. So if the Farmer produces $100 worth of food, then his income will also be $100 since all of this food is bought out and the profits go to the Farmer.

The idea is, of course, to be able to figure out how we should price the goods in order for the citizens of Matrixville to survive; i.e., we must find PF, PT, Pc, PCM, and PSB subject to the following conditions: 
0.25PF + 0.15PT + 0.25Pc - 0.18PCM + 0.20PSB = 0.15PF + 0.28PT - 0.18Pc - 0.17PCM - 0.05PSB = 0.22PF + 0.19PT - 0.22Pc + 0.22PCM - 0.10PSB = 0.20PF + 0.15PT + 0.20Pc + 0.28PCM - 0.10PSB = 0.18PF + 0.23PT - 0.18Pc - 0.17PCM - 0.50PSB = PSR

Exercise 2.5: Explain where this system of equations came from and what it means. (Think about what the left-hand side and the right-hand side of each equation mean.)

Let us denote the column vector (PF, PT, Pc, PCM, PSB) by P, and let E be the coefficient matrix of the above system. We can now rewrite that system as:
EP = P
equivalently
EP - P = (E - I)P = 0
where I is the 5x5 matrix with 1's on the diagonal and 0's everywhere else. The property of the matrix we are using is that Iv = 0 for any vector.

Added by Ronnie A.

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