We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Show that there exists a matrix $B = \left(\begin{array}{ll} B_{1} & B_{2} \\ B_{3} & B_{4} \end{array}\right)$ with $B_{1} \in \mathcal{M}_{n}(\mathbb{R}), B_{2} \in \mathcal{M}_{n,1}(\mathbb{R}), B_{3} \in \mathcal{M}_{1,n}(\mathbb{R})$, $B_{4} \in \mathcal{M}_{1}(\mathbb{R})$ such that: $A_{N} B = \left(\begin{array}{cc} I_{n} & 0 \\ N^{\top} A^{-1} & -N^{\top} A^{-1} N \end{array}\right)$.
We assume $n \geq 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Show that there exists a matrix $B = \left(\begin{array}{ll} B_{1} & B_{2} \\ B_{3} & B_{4} \end{array}\right)$ with $B_{1} \in \mathcal{M}_{n}(\mathbb{R}), B_{2} \in \mathcal{M}_{n,1}(\mathbb{R}), B_{3} \in \mathcal{M}_{1,n}(\mathbb{R})$, $B_{4} \in \mathcal{M}_{1}(\mathbb{R})$ such that: $A_{N} B = \left(\begin{array}{cc} I_{n} & 0 \\ N^{\top} A^{-1} & -N^{\top} A^{-1} N \end{array}\right)$.