>[!info] Symmetry >$A^{\intercal}= A$ ### Determinant $\huge \det A = -1 \iff A \in M_{2\times 2}, \Rn2$ $\huge \det A = 1 \iff A \in M_{3\times 3}, \Rn3$ Reflections will switch *parity* ### Inverse $\huge A^{-1} = A$ ## Vector Form Through a line in vector form ($\Rn2, \Rn3$). $\huge A = -I + \frac{2}{\norms{\vec v}^{2}}\vec v v ^{\intercal} = -I + 2\hat v \hat v ^{\intercal} $ ## Normal Form Through a line in $\Rn2$ or a plane in $\Rn3$. $\huge A = I - \frac{2}{\norms{\vec n}^{2}} \vec n n^\intercal = I - 2 \hat n \hat n^\intercal $