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Mathematics for Economics 2 Degree in Economics Examination of July 1, 2025 [ Last name First name MODULE 1: LINEAR ALGEBRA QUESTION 1 (3 points) Given a system of n linear equations in n unknowns, recall the statement of Cramer’s Theorem and the associated Cramer’s rule. In case n = 3, give an explicit example of a system that has not a unique solution. QUESTION 2 (4 points) Consider the subset of R3 defined by setting: V= {(2a-b,2b- a,a)|a,b € R} (i) Prove that V is a vector subspace of R3. (i) Check whether 9 = {(2,-1,1),(-1,2,0)} is a basis for V. QUESTION 3 (4 points) Consider the map 7 : R? — R$ defined by setting: 2x1 — T2 n T(e)=|zx2-x,1| foreveryz= fa] TI (i) Prove that T is a linear transformation. (i) Check whether T is surjective and give an explicit example of a nonzero vector in the image of T. MODULE 2: DIFFERENTIAL CALCULUS AND OPTIMIZATION QUESTION 1 (3 points) Let g : R° — R and let be R. Under what conditions can we say that the solutions of the equation g(x,y) = db represent a regular constraint? Define the bordered Hessian matrix associated with the constrained optimization problem in which f : R? > R is a twice-differentiable function on R? representing the objective function, and g : R° + R is also twice differentiable and defines the regular constraint g(r,y) = bd. QUESTION 2 (4 points) Determine the critical points of: S(w,y,2) = 28 +99 +22 — yz-30+4 and study their local nature. QUESTION 3 (4 points) Consider the constrained optimization problem: Opt.zg x — 59? sub. x°+y2=1 (i) Can we affirm that there exists (at least) one global maximum point and one global minimum point? Justify your answer. (i) If the answer to the previous point is affirmative, determine the points of global maximum and global minimum. MODULE 3: FINANCIAL CALCULUS QUESTION 1 (3 points) (i) Ilustrate the notion of Net Present Value (NPV) of a financial operation and its use as a criterion of choice among financial operations. (i) Ilustrate the notion of duration and its use as a measure of volatility of the price of a coupon bond. QUESTION 2 (4 points) A machinery with price 10000 € is purchased paying an immediate amount equal to 10% of its price and 3 semi-annual posticipated instalments. The effective compound annual contract interest rate is 10, 25%. Determine the instalments if their profile is described by the vector [l 1 3], calculate the sum of interests and fill in the amortization schedule of the operation. QUESTION 3 (4 points) Consider the following prices of zero-coupon bonds with annual maturities s = 1; 2; 3: v°(0,1) =0,99 0° (0,2) = 0,97 0° (0,3) = 0,94 Assuming that a bond issued at t = 0 that pays annual coupons at 4% and has maturity at # = 4, with reimboursement value, equal to the nominal value, of 100, has non-arbitrage price P(0) at t = 0 equal to 107,28 determine: (i) the price v°(0, 4) and the term structure of spot interest rates h°(0, s), for s = 1; 2; 3; 4; (ii) if the effective rate of return of the bond with coupons at t = 0 is larger than 1,8%. 
