According to the definition of an eigenvalue it satisfies the equation
$Ax=\lambda x$ where $A\in M_{n\times n}^{\mathbb{F}}$.
So that we could have either:
$(A-\lambda I)x=0$ or $(\lambda I-A)x=0$
such that the characteristic equation is either
$det(A-\lambda I)=0$ or $det(\lambda I-A)=0$.
What practical differences result from the two possible definitions?