Proving determinant for orthogonal matrices

Question

If there are two matrices $\det(A)$ and $\det(B)$ such that $\det(A)+det(B)=0$, where both the matrices are real orthogonal matrices.

How can I say the following?

$\det(A+B) = \det(A^T(A+B)B^T)$

Is it a simple property of orthogonal matrices?

Answer 1

$A$ is orthogonal if $AA^T=A^TA=I$. So \begin{align*} \det(A^T(A+B)B^T)&=\det(A^TAB^T+A^TBB^T)\\ &=\det(IB^T+A^TI)\\ &=\det(B^T+A^T)\\ &=\det(B+A)^T. \end{align*} But the determinant of a matrix and it's transpose are equal, so this is also equal to $\det(A+B)$.