So, let $A\in \mathbb{M}_n$. I know that (without prooving it yet :D) - $\det A=\left\{ \begin{matrix}0 & n \text{ is odd}\\ 2 & n \text{ is even} \end{matrix}\right.$
$A =\begin{pmatrix} 1& 0& .& .& .& .& 0& 1\\ 1& 1& 0& .& .& .& .& 0\\ 0& 1& 1& 0& .& .& .& .\\ .& .& .& .& .& .& .& .\\ .& .& .& .& .& .& .& .\\ .& .& .& .& .& .& .& .\\ 0& .& .& .& .& 1& 1& 0\\ 0& .& .& .& .& 0& 1& 1 \end{pmatrix}$
By calculating $\det A$ using first column, I get $\det A=|A_{11}^M|-|A_{21}^M|$. I know that $|A_{11}^M|=1$ for every $n$, and that $|A_{21}^M|=1$ when n is odd, and $|A_{21}^M|=-1$ when n is even.
So far everything is clear. How do I put all of this into a proper proof?