Prove that two block matrices over $\mathbb{F}$ are similar 
Let $\mathbb{F}$ be a field, $n\in\mathbb{N}_{\geq 1}$ and $A\in M_{2n}(\mathbb{F})$, such that $$A=\begin{pmatrix} 0_n & 0_n \\ B & 0_n \end{pmatrix}$$ with $B\in GL_n(\mathbb{F})$. Show that A is similar to the matrix $$\begin{pmatrix} C & 0_2 & \ldots & 0_2 \\ 0_2 & C & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0_2 \\ 0_2 & \ldots & 0_2 & C \end{pmatrix}$$ where $C=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\in M_2(\mathbb{F})$.

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Denote by $\Gamma$ the matrix $$\begin{pmatrix} C&0_2&0_2&\dots&0_2\\0_2&C&0_2&\dots&0_2\\0_2&0_2&C&\dots&0_2\\\vdots&\vdots&\vdots&\ddots&\vdots\\0_2&0_2&0_2&\dots&C\end{pmatrix}.$$
Let $P$ denote the permutation matrix associated to the permutation $\sigma\in \mathcal{S}_{2n}$ such that
$$\sigma(k)=\left\{\begin{array}{ll}
n+\frac{k}{2}&\text{if}\ k\ \text{is even},\\
\frac{k+1}{2}&\text{if}\ k\ \text{is odd}.
\end{array}\right.$$
i.e.,  $P=[p_{i,j}]$, where
$$p_{i,j}=\delta_{\sigma(i),j}.$$
Observe that $P^{-1}\Gamma P$ is given by
$$J=\begin{pmatrix}0_n &0_n\\ I_n&0_n\end{pmatrix}.$$
Next, we know that
$$\begin{pmatrix}I_n&0_n\\0_n&B\end{pmatrix}\begin{pmatrix}0_n&0_n\\I_n&0_n\end{pmatrix}\begin{pmatrix}I_n&0_n\\0_n&B^{-1}\end{pmatrix}=\begin{pmatrix}0_n&0_n\\B&0_n\end{pmatrix}=A.$$
Thus, if $M$ denotes $\begin{pmatrix}I_n&0_n\\0_n&B\end{pmatrix}$, then $$A=MJM^{-1}=M(P^{-1}\Gamma P)M^{-1}=(MP^{-1})\Gamma(MP^{-1})^{-1}.$$
