# Characteristic polynomial for this matrix

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?

• Do a cofactor expansion method on the last column. – dezdichado Apr 22 '18 at 20:00
• @dezdichado You mean if matrix I wrote is $A$, then to find determinant of $tI-A$, I should focus on last column of $tI-A$? – Silent Apr 22 '18 at 20:03
• This is a companion matrix: en.wikipedia.org/wiki/Companion_matrix – Lord Shark the Unknown Apr 22 '18 at 20:04
• The characteristic polynomial is $P(\lambda)=a_0+a_1\lambda+\cdots+a_n\lambda^n$ – marwalix Apr 22 '18 at 20:05
• @Silent, yes that should do it. – dezdichado Apr 22 '18 at 20:05