det(-A) = -det(A) for Odd Square Matrix
The proof for this is straightforward, but I didn’t find it explicitly stated elsewhere on the web – it may just be one of those propositions that’s hard to phrase as a search query. In words: the negative determinant of an odd square matrix is the determinant of the negative matrix.
If is an square matrix and is odd, then .
Negating all elements of a row of the matrix negates the determinant of the matrix (proof – think about it as multiplying a row by the scalar ).
is equivalent to with each of its rows negated. Because is odd:
This fact was needed to prove that if is a special orthogonal matrix and is odd, then has at least one eigenvector with eigenvalue .
Note that if is even, we can prove that using the same technique.