Chapter 7.3, Problem 52E

Find the characteristic polynomial of the $n\times n$ matrix
   $A=\left[\begin{array}{cccccc}0& 0& 0& \dots & 0& a_0\\ 1& 0& 0& \dots & 0& a_1\\ 0& 1& 0& \dots & 0& a_2\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & 0& a_{n-2}\\ 0& 0& 0& \dots & 1& a_{n-1}\end{array}\right]$
Note that the ith column of A is ${\stackrel{\to }{e}}_{i+1}$ , for $i=1,\mathrm{...},n-1$ , while the last column has the arbitrary entries $a_0,\mathrm{...},a_{n-1}$ . See Exercise 51 for the special case $n=3$ .