If $X$ is an $n \times m$ matrix over a field, show that there are invertible matrices $P$ and $Q$ such that

$$Q^{-1} X P=\left[\begin{array}{cc} I_{r} & 0 \\ 0 & 0 \end{array}\right]$$

for some $0 \leqslant r \leqslant \min (m, n)$, where $I_{r}$ is the identity matrix of dimension $r$.

For a square matrix of the form $A=\left[\begin{array}{cc}B & D \\ 0 & C\end{array}\right]$ with $B$ and $C$ square matrices, prove that $\operatorname{det}(A)=\operatorname{det}(B) \operatorname{det}(C)$.

If $A \in M_{n \times n}(\mathbb{C})$ and $B \in M_{m \times m}(\mathbb{C})$ have no common eigenvalue, show that the linear map

$$\begin{aligned} L: M_{n \times m}(\mathbb{C}) & \longrightarrow M_{n \times m}(\mathbb{C}) \\ X & \longmapsto A X-X B \end{aligned}$$

is injective.