TITLE: Is sum of two orthogonal matrices singular?
QUESTION [11 upvotes]: I am trying to solve following problem. 

Let $A, B \in \mathbb{R}^{n\times n}$ be an orthogonal matrices and $\det(A) = -\det(B)$. How can it be proven that $A+B$ is singular?

I could start with implication:
$\det(A)=-\det(B) \Rightarrow B$ is created from $A$ by swapping two lines or columns. 
But I am not sure if this implication is correct.

REPLY [19 votes]: As $A$ and $B$ are real orthogonal and $\det(A)=-\det(B)$, we have $\det(A)\det(B)=-1$. Hence
$$\det(A+B)=\det\left(A(B^T+A^T)B\right)=-\det(B^T+A^T)=-\det\left((B+A)^T\right)=-\det(B+A)$$
and the assertion follows.