Let $m,n \in \mathbb N$, $k$ a field, $X=(k^{n\times m},+)$, and we consider the groups $GL(n,k)$ and $GL(m,k)$. Let $K:=GL(n,k) \times GL(m,k)$. We define \begin{align*} K\times X &\to X\\ ((A,B),M) &\mapsto AMB^{-1} \end{align*}
Note that this defines an action of $K$ on $X$. With this action we can consider $X \rtimes K$. Let $$ G=\left\{\begin{pmatrix} A & M \\ 0 & B \\ \end{pmatrix} : A \in GL(n,k), B \in GL(m,k), M \in X \right\} \, . $$
Show that $G \cong X \rtimes K$.
I've tried to define the following map \begin{align*} \psi:G &\to X \rtimes K\\ \begin{pmatrix} A & M \\ 0 & B \\ \end{pmatrix} &\mapsto (M,(A,B)) \end{align*}
but this is not a morphism since $$\begin{pmatrix} A & M \\ 0 & B \\ \end{pmatrix}\begin{pmatrix} C & M' \\ 0 & D \\ \end{pmatrix}=\begin{pmatrix} AC & AM'+MD \\ 0 & BD \\ \end{pmatrix},$$ so $$\psi\left(\begin{pmatrix} A & M \\ 0 & B \\ \end{pmatrix}\begin{pmatrix} C & M' \\ 0 & D \\ \end{pmatrix}\right)=(AM'+MD,(AC,BD)),$$ and $$\psi\left(\begin{pmatrix} A & M \\ 0 & B \\ \end{pmatrix}\right)\psi\left(\begin{pmatrix} C & M' \\ 0 & D \\ \end{pmatrix}\right)=(M,(A,B)).(M',(C,D))=(M+AM'B^{-1},(AC,BD))$$
Could someone suggest me an appropriate isomorphism between these two groups? Thanks in advance.
First we identify the subgroups of $G$ corresponding to $X$ and $K$: \begin{align*} X &\cong \left\{ \begin{pmatrix} 1 & M\\ 0 & 1 \end{pmatrix} : M \in X \right\} =:\tilde{X}\\ \quad K &\cong \left\{ \begin{pmatrix} A & 0\\ 0 & B \end{pmatrix} : A \in GL_n(k), B \in GL_m(k)\right\} =: \tilde{K} \end{align*} where $1$ is the identity matrix of the appropriate size.
Note that \begin{align*} \begin{pmatrix} A & M\\ 0 & B \end{pmatrix} = \begin{pmatrix} 1 & M B^{-1}\\ 0 & 1 \end{pmatrix} \begin{pmatrix} A & 0\\ 0 & B \end{pmatrix} \in \tilde{X} \tilde{K} \, . \end{align*} This suggests the map \begin{align*} \varphi : G & \to X \rtimes K\\ \begin{pmatrix} A & M\\ 0 & B \end{pmatrix} & \mapsto (MB^{-1}, (A,B)) \, . \end{align*} Can you show that this map is a homomorphism?