Let $A$ and $B$ are orthogonal matrices and $\operatorname{det} A+\operatorname{det} B=0$. Then

Question

Let $A$ and $B$ are orthogonal matrices and $\operatorname{det} A+\operatorname{det} B=0$. Then, $A+B$ is singular, $A+B$ is non-singular, $A+B$ is orthogonal, $A+B$ is skew symmetric

MARKS App · Accepted Answer

Hint: $A A^{\top}=I$ and $B B^{\top}=I$ $\Rightarrow \operatorname{det}\left(A A^{\top}\right)=1$ and $\operatorname{det}\left(B B^{\top}\right)=1$ $\operatorname{det}(A)=-\operatorname{det}(B)$ (Given) $ \Rightarrow(\operatorname{det}(A))^2=1 \Rightarrow(\operatorname{det}(B))^2=1$ $\therefore \operatorname{det}(A+B)=\operatorname{det}\left(A\left(B^{\top}+A^{\top}\right) B\right)$ $=-\operatorname{det}\left(B^{\top}+A^{\top}\right) \quad[\because(\operatorname{det}(A))(\operatorname{det}(B))=-1]$ as $\operatorname{det}(A)=-\operatorname{det}(B)$ $=-\operatorname{det}(B+A)^{\top}$ $=-\operatorname{det}(A+B)^{\top}$ $=-\operatorname{det}(A+B)$ $\Rightarrow 2 \operatorname{det}(A+B)=0$ $\Rightarrow \operatorname{det}(A+B)=0$ $\therefore(A+B)$ is singular

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