= Characteristic polynomial of a companion matrix
{title2=$\det(tI-C(f))=f(t)$}
For a <monic polynomial> $f(t)=t^d+c_{d-1}t^{d-1}+\cdots+c_0$, use the <companion matrix> convention with ones on the subdiagonal and last column $(-c_0,\ldots,-c_{d-1})^{\mathsf T}$. In $tI-C(f)$, the cofactors of the last-column entries, from top to bottom, are $1,t,\ldots,t^{d-1}$. This follows by expanding their bidiagonal minors: the subdiagonal factors $-1$ cancel the cofactor signs. <Laplace expansion> along that column therefore gives $c_0+c_1t+\cdots+c_{d-2}t^{d-2}+(t+c_{d-1})t^{d-1}=f(t)$. The degree-one case is the same identity for the one-by-one matrix $(-c_0)$.
Back to article page
