Definition 3

Let $M$ be a finite-dimensional free $A$-module and $u$ an endomorphism of $M$. The determinant of the endomorphism $X - \bar{u}$ of the free $A[X]$-module $M[X]$ is called the characteristic polynomial of $u$ and is denoted by $\chi_u(X)$.

If $M$ is of rank $n$, it follows from (49) that

for $\det(X - Y \bar{u}) = \det(X . I_n + Y U)$ and $\det(X - \bar{u}) = \det(X . I_n - U)$. It is therefore seen that $\chi_u(X)$ is a monic polynomial of degree $n$, in which the coefficient of $X^{n-1}$ is $-\operatorname{Tr}(u)$ and the constant term is $(-1)^n \det(u)$.