Page 22 - Determinants and Their Applications in Mathematical Physics

2

A Summary of Basic Determinant
Theory

2.1 Introduction

This chapter consists entirely of a summary of basic determinant theory, a
prerequisite for the understanding of later chapters. It is assumed that the
reader is familiar with these relations, although not necessarily with the
notation used to describe them, and few proofs are given. If further proofs
are required, they can be found in numerous undergraduate textbooks.
Several of the relations, including Cramer’s formula and the formula for
the derivative of a determinant, are expressed in terms of column vec-
tors, a notation which is invaluable in the description of several analytical
processes.

2.2 Row and Column Vectors

Let row i (the ith row) and column j (the jth column) of the determinant
A n = |a ij | n be denoted by the boldface symbols R i and C j respectively:

R i = a i1 a i2 a i3 ··· a in ,

(2.2.1)
T
C j = a 1j a 2j a 3j ··· a nj
where T denotes the transpose. We may now write