21 Companion matrix

\det\begin{bmatrix}\lambda&-1&0&\dots\\ 0&\lambda&-1&\ddots\\ \vdots&\vdots&\ddots&\ddots\\ a_{0}&a_{1}&\dots&\lambda+a_{n-1}\end{bmatrix}=

\lambda\det\begin{bmatrix}\lambda&-1&0&\dots\\ 0&\lambda&-1&\ddots\\ \vdots&\vdots&\ddots&\ddots\\ a_{1}&a_{2}&\dots&\lambda+a_{n-1}\end{bmatrix}+(-1)^{n-1}a_{0}\det\begin{% bmatrix}-1&0&0&\dots\\ \lambda&-1&0&\ddots\\ 0&\ddots&\ddots&\ddots\\ 0&\dots&\lambda&-1\end{bmatrix}

so we use the induction hypothesis to deal with the first determinant, and observe that the second is lower triangular, so by Prop 18(ii)

\det(\lambda I-A)=\lambda(\lambda^{n-1}+a_{n-1}\lambda^{n-2}+\dots+a_{2}% \lambda+a_{1})+a_{0}

=\lambda^{n}+a_{n-1}\lambda^{n-1}+\dots+a_{1}\lambda+a_{0}.