I want to show that $a(x) = \det[C(a)-x I]$ where $a(x)=a_0+ax+...+a_{r-1}x^{r-1}+x^r$ and $C(a)$ is the companion matrix:
$$\begin{vmatrix} 0&1&0&\dots& 0 \\ 0&0&1&\dots& 0 \\ 0&0&0&\dots& 0 \\ \vdots&&&&\vdots\\ 0&0&0&\dots& 1\\ -a_0&-a_1&-a_2&\dots&-a_{r-1} \end{vmatrix}$$
I've tried Laplace on this, and I sense that I need to reduce the final matrix in some triangular form, I'm little rusty in matrix algebra. Some hints?
