We assume $\mathbb{K} = \mathbb{R}$. Let $j : \mathcal { L } \left( \mathbb { R } ^ { n } \right) \longrightarrow \mathcal { M } _ { n } ( \mathbb { R } )$ be the linear isomorphism that associates to every endomorphism its matrix in the canonical basis of $\mathbb { R } ^ { n }$. We denote by $O _ { r , s } : = j \left( O \left( Q _ { r , s } \right) \right)$ the subset of matrices associated to the orthogonal group $O \left( Q _ { r , s } \right)$ of $Q _ { r , s }$.
Let $f : \mathbb { R } ^ { n } \rightarrow \mathbb { R } ^ { n }$ be a linear map and $M = j ( f )$ its matrix in the canonical basis of $\mathbb { R } ^ { n }$. Prove that $M \in O _ { r , s }$ if and only if ${ } ^ { t } M I _ { r , s } M = I _ { r , s }$ where $I _ { r , s }$ is the matrix $$I _ { r , s } = \left[ \begin{array} { c c } I _ { r } & 0 _ { r , s } \\ 0 _ { s , r } & - I _ { s } \end{array} \right]$$ $I _ { p }$ denotes the identity matrix of size $p \times p$ and $0 _ { p , q }$ the zero matrix of size $p \times q$ for all integers $p$ and $q$.
What can be said about the determinant $\operatorname { det } ( M )$ of $M$ if $M \in O _ { r , s }$ ?