tailieunhanh - Basic Mathematics for Economists - Rosser - Chapter 15

15 Matrix algebra Formulate multi-variable economic models in matrix format. Add and subtract matrices. Multiply matrices by a scalar value and by another matrix. Calculate determinants and cofactors. Derive the inverse of a matrix. Use the matrix inverse to solve a system of simultaneous equations | 15 Matrix algebra Learning objectives After completing this chapter students should be able to Formulate multi-variable economic models in matrix format. Add and subtract matrices. Multiply matrices by a scalar value and by another matrix. Calculate determinants and cofactors. Derive the inverse of a matrix. Use the matrix inverse to solve a system of simultaneous equations both manually and using a spreadsheet. Derive the Hessian matrix of second-order derivatives and use it to check the second-order conditions in an unconstrained optimization problem. Derive the bordered Hessian matrix and use it to check the second-order conditions in a constrained optimization problem. Introduction to matrices and vectors Suppose that you are responsible for hiring cars for your company s staff to use. The weekly hire rates for the five different sizes of car that are available are Compact 139 Intermediate 160 Large 205 7-Seater People Carrier 340 and Luxury limousine 430. For next week you know that your car hire requirements will be 4 Compact 3 Intermediate 12 Large 2 People Carrier and 1 Luxury limousine. How would you work out the total car hire bill If you worked out total expenditure as 4 x 139 3 x 160 12 x 205 2 x 340 1 x 430 4 606 then you would be correct. You would have also already done a matrix multiplication problem although you may not have realized it Before we look at the formal theory of matrices let us continue with this example for a while longer. If you know that your car hire requirements will change from week to week it can help make calculations clearer if the number of cars required in each category are set out in tabular form as in Table . 1993 2003 Mike Rosser Table Cars required Week 1 Week 2 Week 3 Compact 4 7 2 Intermediate 3 5 5 Large 12 9 5 People carrier 2 1 3 Luxury limousine 1 1 2 The total car hire bill for each week can then be calculated by multiplying the number of cars to be hired in each category by the corresponding price.