hello ,please help me with my math lab question : READ:Now, let p F , p T , p C

Question

hello ,please help me with my math lab question :

READ:Now, let pF, pT, pC, pCM, 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 should we 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 = pF
0.15pF + 0.28pT + 0.18pC + 0.17pCM + 0.05pSB = pT
0.22pF + 0.19pT + 0.22pC + 0.22pCM + 0.10pSB = pC
0.20pF + 0.15pT + 0.20pC + 0.28pCM + 0.15pSB = pCM
0.18pF + 0.23pT + 0.15pC + 0.15pCM + 0.50pSB = pSB

QUESTION:? 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.)

Explanation / Answer

Let denite column vector (PF,PT,PC,PCM,PSB)T by p
and also C to be coefficient matrix of above system. So we can express this system as :
CP = P
or
CP-P = CP-IP = (C-I)P = 0
Where I is the 5x5 matrix eith 1's on the diagonal and 0's everywhere else. The property of the
matrix I we are using is that IV = v for any vector v.

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