So here is the question; it is to prove
$$\det\begin{bmatrix}1&1&0&0&\cdots& 0&0\\0&1&1&0&\cdots& 0&0\\0&0&1&1&\cdots& 0&0\\\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&0&0&\cdots& 1&1\\1&0&0&0&\cdots& 0&1\end{bmatrix}=1+(-1)^n(-1)$$ and the matrix is $n \times n$, and $n\ge2$.
Ok so I started from the first column and it becomes $$\det\begin{bmatrix}1&1&0&\cdots& 0&0\\0&1&1&\cdots& 0&0\\\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&0&\cdots &1&1\\0&0&0&\cdots& 0&1\end{bmatrix}\quad-\text{ or }+\quad \det\begin{bmatrix}1&0&0&\cdots &0&0\\1&1&0&\cdots& 0&0\\0&1&1&\cdots& 0&0\\\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&0&\cdots& 1&1\end{bmatrix}$$ Then I had difficulty to proceed, someone please help?