Page 148 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

The following computation illustrates the introduction of a row of zeros when there are two proportional rows:
Each of the following matrices has two proportional rows or columns; thus, each has a determinant of zero.

Evaluating Determinants by Row Reduction

We shall now give a method for evaluating determinants that involves substantially less computation than the cofactor expansion
method. The idea of the method is to reduce the given matrix to upper triangular form by elementary row operations, then compute the
determinant of the upper triangular matrix (an easy computation), and then relate that determinant to that of the original matrix. Here
is an example:

EXAMPLE 4 Using Row Reduction to Evaluate a Determinant

Evaluate where

Solution

We will reduce A to row-echelon form (which is upper triangular) and apply Theorem 2.2.3:

Remark The method of row reduction is well suited for computer evaluation of determinants because it is computationally efficient