My homework is due tomorrow (12h left), that means I've already lost, but I'm looking genuinely for a possible solution.
The Problem:
Let $n \in \mathbb{N}$ and $1 \leq r \leq n$. Let $A = [a_{ij}]_{i,j}$ be an $r \times r$ matrix, let $B = [b_{ij}]_{i,j}$ be an $r \times (n-r)$ - matrix, and let $C = [c_{ij}]_{i,j}$ be an $(n - r) \times (n-r)$ matrix. We define an $n \times n$ matrix
$$D = [d_{ij}]_{i,j} := \left[\begin{array}{c|c} A & B \\ \hline 0 & C \end{array}\right].$$
Show that: $det(D) = det(A) \cdot det(C)$.
Further, distinguish between the cases when $C$ is invertible and when $C$ is not invertible.
To be sincere: I don't even have a solution approach.
