T = C M F P A 20 17 25 .20 10 .20 .10 .30 0 .25 05 10 20 10 15 .10 .28 .40 .20 0 15 .15 .80 C M F P A 40 .15 Notice that

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T C M F P A 20 17 25 20 10 20 10 30 0 25 05 10 20 10 15 10 28 40 20 0 15 15 80 C M F P A 40 15 Notice That 1 (73.89 KiB) Viewed 17 times

T = C M F P A 20 17 25 .20 10 .20 .10 .30 0 .25 05 10 20 10 15 .10 .28 .40 .20 0 15 .15 .80 C M F P A 40 .15 Notice that each column of T sums to one, indicating that all output of each sector is distributed among the five sectors, as should be the case in an exchange economy. The system of equations Tr = x must be satisfied for the economy to be in equilibrium. As you saw above, this is equivalent to the system B.x = 0. (1) Write out the five equations in the equation Tr = r. (2) Obtain a homogeneous linear system Br = 0 equivalent to Tr = r. What is B? Hint: Collect the like terms after moving all non-zero terms to LHS. (3) Solve Br = 0 using any kind of calculator or computer programming. Specify what calculator or computing language you used. (4) Reduce augmented matrix [B10] to RREF form step by step. You may use either hand-computation or programming such as Matlab or python. Attach codes at the end of the report if you used programming. A reference to python coding for Gaussian elimination is HERE, and video is HERE. (5) Write the general solution of Bx = 0. (6) Suppose that the economy described above is in equilibrium and A = 100 million dollars. Calculate the values of the outputs of the other sectors. (7) As already observed, each column of T sums to one. Consider how you obtained B from T and explain why each column of B must sum to zero. (8) Let B be any matrix of any shape, with the property that each column of B sums to zero. Explain why the reduced echelon form of B must have a row of zeros.