9.13 Use Crammer s rule to solve the following two systems of equations.

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1 Curtis Kephart Econ 2B Mathematics for Economists Problem Set 2 Problem 9. Use Crammer s Rule to Invert the following 3 matrices. a) 4 3 3, b) 5 6, 8 The (very long) method of computing adj A for a 3x3 matrix is found on pg 95, (Simon Blume) , / / / / / / / / / Note that the back of the book as field a 32 as -2/37, however, I think it s wrong. That field is (a *a 32-a 3*a 2), this is (*-*2)=-2. And for the adj at C 23, you need to put a negative in front of the -2, leaving 2. c), 9.3 Use Crammer s rule to solve the following two systems of equations. a) : ,,

2 7 /7 / /7 5/7, /7 /7, 2/7 5/ b) : 4 6 7,,, , Verify the conclusions of Theorem 9.5 for the following pairs of matrices. a) 4 5 4, 3 ), 4 5, det 4 5,det 4 5 2) det det ) det

3 b) 4 5, ), ) det I add the ( ) because with the zero multiplier, the rest is irrelevant. 3) det det c), ),

4 2) det det 3) det det, Problem , 4, 6, a) show that det(a+b) deta + detb. det b) Show that deta +detb = detc and relate this to fact 26.6 we ve already established that: , they match!

5 Regarding fact 26.6, it says that if matrix A & B are nxn matrices which differ only in one row (their i th row). And let matrix C be an nxn matrix, whose i th row be the sum of the i th rows from A and B, then: Clearly, matrix C is a matrix whose first row is the sum of matrix A & B s first row. And all three matrices have their remaining rows (with these, just one extra row) the same. Thus question 26.9 illustrates fact a) for the following three matrices, apply row operations to obtain an upper-triangle matrix, and then use fact 26. (pg 7), to find the determinant. i) b) for i) Because DetA (or A4 in this case) is non-zero (-6 in this case), the matrix is nonsingular it has a matrix inverse. ii)

6 Which really is the determinant. b) for ii), because the determinate is non-zero, the matrix is non-singular it has a matrix inverse. iii) DetAi = 2*3*-4*9 = -456 B for iii) once again, because DetA is not zero, it means the matrix is nonsingular. Problem 3 Arbitrage Pricing a) Because assets a & b are closely related to the stock market, they likely represent the stock market itself. Though, for asset b which pays off if the Dow increases or remains the same I m at a loss which type of real-world asset that might be. Perhaps we re assuming asset b is a stock which pays dividends (blue chip), and the stocks in asset a do not ( growth stocks ). Asset c, which pays off only if the stock market is down, probably represents bonds, or at least government bonds. However, I thought that inverse correlation has broken down over the years.

7 b) &,, det. You want x2 and x3. Costs equal,, c) Suppose there is a fourth asset on the market, namely a riskless asset which pays. Suppose, 3, & 3.5. Describe how you can make limitless profits regardless of what happens to the DJIA by an appropriate choice of a portfolio x arbitrage., 3 4. Yet there s a new asset, a4, at cost p4=3.5 that delivers the same payoff. I d buy at 3.5 and sell at 4 to all who d be willing to buy, thereby profiting.5 per transaction. d) Clearly, p4 must 4 for there to exists an arbitrage opportunity. So if p4=4, there s no arbitrage to take advantage of. e) Suppose that prices in the market adjust (reflecting reality) so that arbitrage becomes impossible. That is, if it is extremely profitable to sell asset i, then many people attempt to sell it, driving its price down. Suppose p=, p2=3, p3=, and suppose there is an asset 5 which pays off at /2 And I guess we re forgetting about a 4. What must the price of asset 5 be, once the market adjusts to eliminate all opportunities for arbitrage? Assuming we re still dealing with the same,,, we need to find a combination of them that results in /2, and find that price /2 Using, /2 /2. With the prices for,, above, total price for this portfolio is 3

8 Thus 3 for there to exist no arbitrage opportunities. f) Suppose that only the following four assets exist: assets 2, 3, 4 (described above) 2 and asset 6, which pays off at. Suppose, as in (e), that prices adjust so that 2 arbitrage becomes impossible. Is it possible to calculate p6 if you only know p2, p3 and p4. Without having to do any calculations, it is clear that a2, a3 and a4 do not offer the portfolio combinations necessary to synthetically recreate a6, & thereby find a6 s price. When trying to solve the new, I find that matrix has a determinant of zero, meaning it is not invertable. Problem 4 4., Compute all the partial derivatives of the following functions: , 2 3, 2 3 2, 2 3, 2 3 3

9 Using the quotient rule.,,,,,, & Compute the partial derivatives of the Cobb-Doublas production function Now I m told I should have solved this for δq/δx_i which is the following: And of the Constant elasticity of Substitution (CES) production function,