Linear Algebra, 5th Edition

4.3 Properties of Determinants

In Theorem 3.1, we saw that performing an elementary row operation on a matrix can be accomplished by multiplying the matrix by an elementary matrix. This result is very useful in studying the effects on the determinant of applying a sequence of elementary row operations. Because the determinant of the $n\times n$ identity matrix is 1 (see Example 4 in Section 4.2), we can interpret the statements on page 217 as the following facts about the determinants of elementary matrices.

1. (a) If E is an elementary matrix obtained by interchanging any two rows of I, then $\text{det}(E)=-1$.

2. (b) If E is an elementary matrix obtained by multiplying some row of I by the nonzero scalar k, then $\text{det}(E)=k$.

3. (c) If E is an elementary matrix obtained ...