Page 128 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

2.1 As noted in the introduction to this chapter, a “determinant” is a certain kind
of function that associates a real number with a square matrix. In this
DETERMINANTS BY section we will define this function. As a consequence of our work here, we
COFACTOR EXPANSION will obtain a formula for the inverse of an invertible matrix as well as a
formula for the solution to certain systems of linear equations in terms of
determinants.

Recall from Theorem 1.4.5 that the matrix

is invertible if . The expression occurs so frequently in mathematics that it has a name; it is called the

determinant of the matrix and is denoted by the symbol or . With this notation, the formula for given in

Theorem 1.4.5 is

One of the goals of this chapter is to obtain analogs of this formula to square matrices of higher order. This will require that
we extend the concept of a determinant to square matrices of all orders.

Minors and Cofactors

There are several ways in which we might proceed. The approach in this section is a recursive approach: It defines the

determinant of an matrix in terms of the determinants of certain matrices. The

matrices that will appear in this definition are submatrices of the original matrix. These submatrices are given a special

name:

DEFINITION

If is a square matrix, then the minor of entry is denoted by and is defined to be the determinant of the

submatrix that remains after the th row and th column are deleted from . The number is denoted by

and is called the cofactor of entry .

EXAMPLE 1 Finding Minors and Cofactors
Let

The minor of entry is