# Show that $p(C)\neq 0$ for a particular $n\times n$ matrix $C$ and a polynomial $p$ of degree less than $n$.

Let $a_0,a_1,...,a_{n-1}$ complex numbers. Let $C=[c_{ij}]$ the $n\times n$ matrix so that $c_{ij}=1$ if $i=j+1$, $c_{jn}=a_{j-1}$ and the others entries are zero. Show that if $p$ is a polynomial of degree less than $n$ then $p(C)\neq 0$.

You can give me hints for show this problem please.

I try using theory of minimal and characteristic polynomial but i not can't find a result for to prove this exercise.

We have $$C=\begin{bmatrix} 0 & 0 & \dots & 0 & a_0 \\ 1 & 0 & \dots & 0 & a_1 \\ 0 & 1 & \dots & 0 & a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & a_{n-1} \end{bmatrix}$$ and so $C$ is the companion matrix of the polynomial $q(t)=-(a_0 + a_1 t + \cdots + a_{n-1}t^{n-1} + t^n)$.
It is well-known that the minimal polynomial of $C$ is equal to $q(t)$.
• @Ihf... which is the method for find the inverse of $C$? – Luis Prado Dec 14 '17 at 1:25