Explain where this system of equations came from and what it means. (Think about

Question

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)T by p, and let C be the coefficient matrix of the above system. We can now rewrite that system as

Cp = p

or equivalently

Cp - p = Cp - Ip = (C - 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 I we are using is that Iv = v for any vector v.

Explanation / Answer

first of fall explanation is asked for these 5 set of equation written below. at the end.

Let us denote the column vector (pF, pT, pC, pCM , pSB)T by p, and let C be the coefficient matrix of the above system. now those equation can be written as Cp=p and it is simplified further above .

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

these equation can be explained simply seeing p as expenditure too as mentioned above. assumong expenditure on food, cloths, furniture etc. these equation has been made. i.e/ left side represents expenditure. right hand side represents total income. i.e. from that much money, exactly these thinngs can be purchased.

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