Let $X \in Mat_p(\mathbb{K})$, $C \in Mat_q(\mathbb{K})$, $D\in Mat_{p,q}(\mathbb{K})$ where $p+q=n$.
Consider the function
$\alpha: \mathbb{K}^p\times...\times\mathbb{K}^p\rightarrow\mathbb{K}$
Defined by
$\alpha(X_1,...,X_p)=det\begin{bmatrix}
X_{p\times p} & D_{p\times q} \\
0_{q\...
You don't need that. But because invertible matrices are dense in the space of all matrices, you can use a continuity argument to deduce it holds for all $X$ if you prove it for invertible $X$.
