60 Conclusion of realization

and so until the cofactor of the entry $s+\alpha_{n-1}$ in place $(n,n)$ is

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

C{\hbox{adj}}(sI-A)B=\begin{bmatrix}\gamma_{0}&\gamma_{1}&\dots&\gamma_{n-1}% \end{bmatrix}\begin{bmatrix}1\\ s\\ \vdots\\ s^{n-2}\\ s^{n-1}\end{bmatrix}

=\gamma_{0}+\gamma_{1}s+\dots+\gamma_{n-1}s^{n-1}.