59 Cofactors

The cofactor of the entry $\alpha_{0}$ in place $(n,1)$ is

(-1)^{n-1}\det\begin{bmatrix}-1&0&\dots&0\\ s&-1&\ddots&0\\ \ddots&\ddots&\ddots&\vdots\\ \vdots&0&s&-1\end{bmatrix}=1

The cofactor of the entry $\alpha_{1}$ in place $(n,2)$ is

(-1)^{n-2}\det\begin{bmatrix}s&0&\dots&0\\ 0&-1&\ddots&0\\ \ddots&\ddots&\ddots&\vdots\\ \vdots&0&0&-1\end{bmatrix}=s