We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$. Show that if $N^{\prime} = \left(\begin{array}{ll} N_{1}^{\prime} & N_{2}^{\prime} \end{array}\right)$ then $$\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = \left(N_{1}^{\prime\top} A_{s} N_{1}^{\prime}\right)\left(N_{2}^{\prime\top} A_{s} N_{2}^{\prime}\right) - \left(N_{1}^{\prime\top} A_{s} N_{2}^{\prime}\right)^{2} + \left(N_{1}^{\prime\top} A_{a} N_{2}^{\prime}\right)^{2}$$
We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that if $N^{\prime} = \left(\begin{array}{ll} N_{1}^{\prime} & N_{2}^{\prime} \end{array}\right)$ then
$$\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = \left(N_{1}^{\prime\top} A_{s} N_{1}^{\prime}\right)\left(N_{2}^{\prime\top} A_{s} N_{2}^{\prime}\right) - \left(N_{1}^{\prime\top} A_{s} N_{2}^{\prime}\right)^{2} + \left(N_{1}^{\prime\top} A_{a} N_{2}^{\prime}\right)^{2}$$