My book says that you form the character equation as $\begin{vmatrix} \lambda I - A \end{vmatrix}$. However I see occasionally on stack exchange and other resources that define the character equation as $\begin{vmatrix}A - \lambda I\end{vmatrix}$. Why does this appear to be such a trivial matter? Here are some examples below where I have found the latter being used.
Geometric multiplicity of repeated Eigenvalues
https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors (under "Eigenvalues and the characteristic polynomial"
The two only differ by a factor of $(-1)^n$, so it's not a big deal. In my opinion, the better convention is $\det (\lambda I - A)$, because that guarantees that the characteristic polynomial is always monic (has leading coefficient $1$). But it is a convention; both conventions are basically fine.