For questions about companion-matrices

The companion matrix of the monic polynomial $p(t)=c_0+c_1 t+\cdots+c_{n-1} t^{n-1}+t^n,$ is the square matrix defined as

$C(p)=\begin{bmatrix}0&0&\cdots&0&-c_0\\1&0&\cdots&0&-c_1\\0&1&\cdots&0&-c_2\\\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&\cdots&1&-c_{n-1}\end{bmatrix}$.

The characteristic polynomial as well as the minimal polynomial of $C(p)$ are equal to $p$.

In this sense, the matrix $C(p)$ is the "companion" of the polynomial $p$.

Companion matrices are too related with the Rational Canonical Form of a matrix.