Page 145 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

2.2 In this section we shall show that the determinant of a square matrix can be
evaluated by reducing the matrix to row-echelon form. This method is important
EVALUATING since it is the most computationally efficient way to find the determinant of a
DETERMINANTS BY ROW general matrix.
REDUCTION

A Basic Theorem

We begin with a fundamental theorem that will lead us to an efficient procedure for evaluating the determinant of a matrix of any
order .

THEOREM 2.2.1

Let be a square matrix. If has a row of zeros or a column of zeros, then .

Proof By Theorem 2.1.1, the determinant of A found by cofactor expansion along the row or column of all zeros is

where , , are the cofactors for that row or column. Hence is zero.
Here is another useful theorem:
THEOREM 2.2.2

Let A be a square matrix. Then .

Proof By Theorem 2.1.1, the determinant of A found by cofactor expansion along its first row is the same as the determinant of
found by cofactor expansion along its first column.

Remark Because of Theorem 2.2.2, nearly every theorem about determinants that contains the word row in its statement is also true
when the word column is substituted for row. To prove a column statement, one need only transpose the matrix in question, to convert
the column statement to a row statement, and then apply the corresponding known result for rows.

Elementary Row Operations

The next theorem shows how an elementary row operation on a matrix affects the value of its determinant.

THEOREM 2.2.3