Square Matrices

Abstract

The essential ingredient for the study of square matrices is the determinant. For reasons that will be given in Section 2.5, as well as in Chapter 6, it is useful to consider matrices with entries in a ring. This allows us to consider matrices with entries in ℤ (rational integers) as well as in K[X] (polynomials with coefficients in K). We shall assume that the ring A of scalars is a commutative (meaning that the multiplication is commutative) integral domain (meaning that it does not have zero divisors: ab = 0 implies either a = 0 or b = 0), with a unit denoted by 1, that is, an element satisfying 1x = x1 = x for every x ∈ A. Observe that the ring Mn(A) is not commutative if n ≥ 2. For instance $\left( {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right) \ne \left( {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right).$

An element a of A is invertible if there exists b ∈ A such that ab = 1. The element b is unique (because A is an integral domain), and one calls it the inverse of a, with the notation b = a^-1. The set of invertible elements of A is a multiplicative group, denoted by A^*. One has (ab)^-1 = b^-1a^-1 = a^-1b^-1.