Standard form for the Characteristic Matrix/Polynomial

Question

I'm currently taking Linear Algebra and Differential Equations, and in talking about eigenvalues of a matrix, both professors have given the same information: for some square n x n matrix A, the eigenvalues of A are given by the roots of the characteristic polynomial. However, the two differ in the definition of the characteristic matrix of A. One gives $\lambda I_n-A$ and the other gives $A-\lambda I_n$. While the two yield the same eigenvalues, is there a standard form we refer to when talking about the characteristic matrix? Does it even matter?

Answer 1

$\text{det}(\lambda I - A)$ seems to be the more common one. It has the advantage of resulting in a polynomial whose leading coefficient is $1$, not $-1$. It was mentioned in the comments that these definitions vary by a sign, which means that this is not very important. The reason they vary by a sign is the alternating multilinearity of determinant.