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 the effect each elementary row operation will have on det A as we reduce A to echelon form. We use here the concise notation

for the determinant of the matrix A.

The following example shows how to lay out the computation of det A by using row operations to reduce A to echelon form.

If the square matrix A has only integer entries, it is always possible to carry out the reduction to echelon form without introducing any fractions. In the computation of determinants (as opposed to the solution of linear systems), we have the additional flexibility of using elementary column operations; their effects are precisely analogous to those of elementary row operations, as described in Theorem 1. In Example 2, we illustrate this by finishing differently the evaluation in Example 1.