Basic matrix algebra for economists
Issues in matrix algebra
Introductory matrix algebra is a familiar component of undergraduate mathematical economics modules. Recently, I have introduced two new elements to my teaching of this topic in a level 2 mathematical economics module. The first of these changes is the introduction of a simple formula, Dodgson's condensation formula, which allows students to gain confidence in manipulating matrices and calculating their determinants. While this has proved very useful, I am unaware of any mathematical economics text in which it is presented. The second development has been the incorporation of Excel to examine topics in matrix algebra. Both of these developments have proved to be useful additions to the module and are outlined below.
Dodgson's condensation formula
The calculation of matrix determinants is a fundamental element of matrix algebra. Typically, analysis will start with the simple case of a (2x2) matrix before (3x3) matrices are introduced to demonstrate Laplace expansion. Indeed, this is the standard format of mathematical economics texts. However, simply providing students with an exercise sheet requiring the calculation of a list of (2x2) and (3x3) matrices has a number of potential drawbacks. First, it can obviously appear dull and repetitive. Second, until the subsequent examples class is attended, the student will not know whether the method followed, and the answers derived, are correct. Finally, such a strategy may not be viewed as having an overall goal or objective. To overcome these problems, I have employed Dodgson's condensation formula as a means of familiarising students with the mechanics of matrix manipulation and the calculation of determinants (it may be of interest to note that the formula was derived by Charles Dodgson, a 19th century Oxford mathematician who was perhaps better known as Lewis Carroll). For a matrix A, Dodgson's condensation formula states that:
| A | . | B | = ( |AFF | . |ALL | ) - ( |AFL | . |ALF | )
| | denotes a determinant,
AFF is A with its first row and first column deleted,
ALL is A with its last row and last column deleted,
AFL is A with the first row and last column deleted,
ALF is A with the last row and first column deleted,
B is A with the first and last rows and columns deleted.
In my experience, the advantages of using this formula in the teaching of basic matrix algebra are numerous. For example, with a (3x3) matrix provided, examination of this formula has the following benefits:
- It requires that a student undertakes some basic manipulation of matrices when calculating the above sub-matrices,
- It results in the calculation of determinants of matrices with different dimensions within a single question,
- It introduces a clear objective (showing that the formula holds) for what might otherwise be viewed an exercise without a purpose,
- Checking whether the LHS and RHS of the formula are equal allows students to see whether they are employing the correct methods.
The use of spreadsheets
A second addition to my current teaching of matrix algebra, is the incorporation of Excel. Spreadsheets have been created to allow the following topics to be automated:
- Matrix inversion
- Dodgson's condensation formula
- The solution of equations via matrix inversion
- The solution of equations via Cramer's Rule
At present the spreadsheets are provided primarily as a means of allowing students to generate their own examples and check their working, particularly when revising for assessments, although they are introduced during formal teaching sessions. However, in future they could be more actively incorporated into lectures or into computer workshops. It may be the case that the ability to construct and manipulate such a spreadsheet is itself the objective of a workshop. At present an additional benefit of providing these spreadsheets is that it provides students with further exposure to the use of I.T. The present topics could also be extended to address applications of matrix algebra or could be modified to demonstrate more basic issues (for example, linear dependence and its impact upon the determinant of a matrix). While the use of Excel to examine matrices is beginning to appear in some texts, I thought others teaching level 2 mathematical economics or related modules might benefit from the provision of ready-made spreadsheets for use in the classroom.
In my experience the use of the Dodgson's condensation formula and Excel have helped to refresh the presentation of what can sometimes be fairly dry, routine material. Informal feedback has proved positive, and I would encourage the adoption and extension of the above developments by others teaching similar modules, particularly as adoption is costless. It may also be noted that the spreadsheets provide a very quick means of generating questions and solutions for examples classes and assessments!
Dodgson, C. (1866) 'Condensation of determinants', Proceedings of the Royal Society of London 15, 150-155. [If your institution has access to the JSTOR project, then you can retrieve the paper through this URL: https://www.jstor.org/stable/112607 ]