The characteristic polynomial of a matrix $A \in \mathbb{C}^{n \times n}$, $p_A (\lambda) = \det(A-\lambda \cdot E)$ can be used to find the eigenvalues of the linear function $\varphi:\mathbb{C}^n \rightarrow \mathbb{C}^n, \varphi(x) := A \cdot x$, as the eigenvalues are the roots of $p_A(\lambda)$. So, for finding the eigenvalues, the sign of the characteristic polynomial isn't important. At the moment, this is to only application of the characteristic polynomial that I know.
Do other applications of the characteristic polynomial exist, where the sign of it is important?
Can I make any statements about the matrix $A$ when I know the sign of its characteristic polynomial?
Best Answer
Actually the characteristic polynomial is often defined as $$ \chi_A=\det(I_nX-A)\in k[X] $$ so as to be always monic (of degree $n$); see for instance in Wikipedia. This differs by a sign (and by calling the identity matrix by the more ususal name of $I_n$) from the definition you cited. The fact that the two contradicting definitions coexist shows that the matter of a factor $(-1)^n$ is not considered of great importance.
In my experience however, in most applications of the characteristic polynomial other than just for searching eigenvalues, the fact that it is monic is of importance. (One such application is to show that certain values are algebraic integers over the base ring, i.e., solutions of a monic polynomial equation.) For sure, in those applications monic-up-to-a-sign will be easily seen to do the job as well, but it is more convenient if the characteristic polynomial is just monic, period.
Also consider the statement "the coefficient of degree $n-i$ of $\chi_A$ is the $i$-th symmetric function of minus the eigenvalues of $A$ (taken with thir algebraic multiplicities)". With the definition you gave, you'd need to throw in another "minus".