Differential Equations and Linear Algebra, 4th Edition

Determinants and Elementary Row Operations

As we described in Section 3.6, the determinant of any square matrix A can be evaluated directly by carrying out a cofactor expansion of det A along any row or column of A. Because this approach involves so much computational labor, it is more efficient instead to reduce A to an echelon matrix R. Because any square echelon matrix is (upper) triangular, the determinant of the echelon matrix R is simply the product of its diagonal elements. But because we have altered the matrix A by transforming it into R, the question is this: What effects do elementary row operations have on the determinant of A?

The following theorem summarizes Properties 1, 2, and 5 of Section 3.6 and tells us how to keep track of ...