# 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 $A$ is an $n \times n$ square matrix and $n$ is odd, then $\det(-A) = -\det(A)$.

## Proof

Negating all elements of a row of the matrix $A$ negates the determinant of the matrix (proof – think about it as multiplying a row by the scalar $-1$).

$-A$ is equivalent to $A$ with each of its rows negated. Because $n$ is odd:

## Context

This fact was needed to prove that if $A$ is a special orthogonal matrix and $n$ is odd, then $A$ has at least one eigenvector with eigenvalue $1$.

Note that if $n$ is even, we can prove that $\det(A) = \det(-A)$ using the same technique.