This question is from Artin's Algebra:
Let $V$ be a vector space with basis $(v_0,\ldots, v_n)$ and let $a_0,\ldots,a_n$ be scalars. Define a linear operator $T$ on $V$ by rules $T(v_i)=v_{i+1}$ if $i<n$ and $T(v_n)=a_0v_0+\ldots+a_nv_n$. Determine the matrix $T$ with respect to the given basis, and the characteristic polynomial of $T$.
I think the required matrix is $$\begin{pmatrix} 0 & 0 & \cdots & 0 &a_0 \\ 1 & 0 & \cdots & 0 &a_1 \\ 0 & 1 & \cdots & 0 &a_2 \\ \vdots & \vdots& \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1& a_n \end{pmatrix}$$ How to compute characterstic polynomial?