For a matrix $A$, Is there any difference to state the characteristic polynomial in terms of $\det(A-\lambda I)$ or $\det(\lambda I - A)$?
Because I need to replace the $\lambda$ with some other values which are not its roots, so I think based on the definition, the results will be different. I would like to know which one is the polynomial characteristic of $A$?
I've seem both of them used as definition, although I think that I've seen far more often the first option. Anyway, they are either equal is one of them is minus the other one. Therefore, they have the same roots (which are the eigenvalues of $A$).