Suppose A and B are two orthogonal matrices of the same size such that det⁡(A)+det⁡(B)=0 then

As A, B are orthogonal matrices, AA′=BB′=I, det⁡(A)≠0,det⁡(B)≠0.  Let  C=A+B C′=A′+B′⇒AC′=AA′+AB′=I+AB′⇒AC′B=B+AB′B=B+ANow, det⁡(A+B)=det⁡AC′B=det⁡(A)det⁡C′det⁡(B) =det⁡(A)det⁡(C)(−det⁡(A))⇒ det⁡(C)=−(det⁡(A))2det⁡(C)=−det⁡(C)⇒ det⁡(C)=0