Swapping last 2 rows of a matrix negates its determinant

Dependencies:

1. Determinant

Let $A$ be an $n$ by $n$ matrix. Let $B$ be the matrix obtained by swapping the last 2 rows of $A$ ($\langle n-1 \rangle \leftrightarrow \langle n \rangle$).

Then $|B| = -|A|$.

Proof

Proof can be found in the book: 'Elementary Linear Algebra', 4th Edition by Stephen Andrilli and David Hecker, in Appendix A, Theorem 3.3, page 648.

Dependency for:

1. Determinant after elementary row operation

Info:

• Depth: 5
• Number of transitive dependencies: 6

Transitive dependencies:

1. Group
2. Ring
3. Semiring
4. Matrix
5. Submatrix
6. Determinant