Let $A$ be a $2 \times 2$ orthogonal matrix such that $\det A = −1$.

Question

Let $A$ be a $2 × 2$ orthogonal matrix such that det $A = −1$. What can you say about the eigenvalues of A. I do know that if det orthogonal matrix is equal to 1/-1 than this is rotation matrices. What about eigenvalues i suppose has value of 1(from experiance) howevere i do not know exactly answer

Answer 1

If $A$ is an orthogonal $2\times2$ matrix and $\det A=-1$, then $A=\bigl(\begin{smallmatrix}a&b\\b&-a\end{smallmatrix}\bigr)$, where $a,b\in\mathbb R$ are such that $a^2+b^2=1$.

Answer 2

Clearly, in this case det($A$)$=-1$ implies that eigenvalues are $1$ and $-1$.

Reason: Let $\alpha \pm i\beta$ be the eigenvalues, where $\alpha,\beta \in\mathbb R$ then product of eigenvalues equals determinant implies $\alpha^2+\beta^2=-1$ (a contradiction)