The Ubiquitous e

Abstract

Texts on elementary matrix theory, e.g. [1], usually mention the following three methods for computation of the determinant of an n by n matrix A = [ai,]: (a) Direct computation using the definition via permutations; (b) reduction by Gaussian elimination; and (c) expansion by minors. If we let a computation mean any addition, subtraction, multiplication or division (excluding those used in the determination of the signs), then it is easy to see that (a) requires n! n I computations. Many texts, e.g. [2], show that (b) requires 2n3/3 n2/2 + 5n/6 I computations. These are used to compare the efficiency of (a) and (b), with the conclusion that (b) requires fewer computations for n >2. Here we wish to derive an explicit formula for the number of computations required by (c). Instead of just stating and proving the result, we will exhibit the process used to derive the formula so that the reader can share in the surprise of discovery. Calculation of the determinant by expansion by the first row uses the equation det A = a,,A + al2A,2+ * * * + alnA ni where the A,, are the appropriate cofactors. If we denote by Nn the number of computations required by method (c), then calculation of a cofactor requires Nn-l computations. Once these are found, there are then n multiplications and n I additions to perform. This yields the recursion formula