duction to Financial Mathematics - Semester 1, 2017 Assignment 6 Algebra To be computer entered between: 9am Friday 28th March and 5pm Monday 1st May Examples: 1 2 1 A. For the matrix A = 0 0 3 1 0 1 (i) calculate det(A) and deduce that A is invertible; (ii) use row operations to calculate A1 ; (iii) evaluate det(A1 ) and verify that det(A1 ) det(A) = 1. B. (Cost allocation model) The company accountant allocates costs by separating costs into direct costs and internal costs for each department. The development department charges 10% of its total monthly costs (which are $x) to the promotional department. The promotional department charges 5% of its total monthly costs (which are $y) to the development department. The direct costs for the development department are $20,400 and for the promotional department they are $9,900. Hence the following matrix equation must hold: \u0014 \u0015 \u0014 \u0015\u0014 \u0015 \u0014 \u0015 20400 0 0.05 x x + = . 9900 0.1 0 y y Find the total costs for each department. Questions: a b c 1. Let A = d e f . Assuming that det(A) = 8, find: g h i (a) det(2A) (b) det(A1 ) (c) det(2A1 ) (d) det[(2A)1 ] a d g a b c d e f (e) det b e h (f) det g h i (g) det a/2 b/2 c/2 c f i d e f g h i 4 0 2 2. For the matrix A = 1 2 1 0 3 1 (a) calculate det(A) and deduce that A is invertible; (b) use row operations to calculate A1 ; (c) evaluate det(A1 ) and verify that det(A1 ) det(A) = 1; (d) use the matrix inverse A1 to solve the system of linear equations: 4x + 2z = 6 , x + 2y + z = 1 , 3y + z = 3 . a 0 0 3. (a) Under what conditions is the matrix C = 0 b 0 invertible? 0 0 c Find C 1 and det(C 1 ). a 0 0 a g h (b) When are the \"triangular\" matrices A = d b 0 and B = 0 b i e f c 0 0 c invertible? 4. A town has four industries: agriculture, fishing, lumber and textiles. For a Leontief open economic model, the input-output matrix for this economy is 0.3 0.1 0.1 0.6 0.1 0.3 0 0.1 A= 0.3 0.4 0.2 0.2 0.1 0.2 0.1 0.1 where the columns represent the inputs of agriculture, fishing, lumber and textiles, respectively. (a) Which industries are profitable? (b) Which industries break even? (c) What inputs does the fishing industry require from all four industries to make $1000 of fishing products? (d) What is the difference between the fishing industry's output and inputs? (e) How much profit does the fishing industry make for each dollar of output? (f) What inputs does the lumber industry require from all four industries to make $1000 of lumber products? (g) What is the difference between the lumber industry's output and inputs? (h) How much profit does the lumber industry make for each dollar of output? 5. The engineering department of a firm charges 20% of its total monthly costs to the computer department and the computer department charges 30% of its total monthly costs to the engineering department. Say the total monthly costs of the engineering department is $x and the total monthly costs of the computer department is $y. Say, during a given month, the engineering department's costs, not including what it paid to the computer department, were $94, 000, and the computer department's costs, not including what it paid to the engineering department, were $112, 000. (a) Write equations for x and y. (b) For x = (x, y) and d = (94000 , 112000), determine the matrix A in the equation x = d + Ax. (c) Calculate the total monthly costs for each department, to the nearest dollar. 6. For the matrix A = \u0014 2 4 6 3 \u0015 find A1 as products of elementary matrices