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.

Proposition

If is an square matrix and is odd, then .

Proof

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:

Context

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.

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