# Standard form for the Characteristic Matrix/Polynomial

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?

• It does not matter as $det(-B) = (-1)^ndet(B)$ for any $n\times n$ matrix $B$. – AnonymousCoward Apr 24 at 13:06

$$\text{det}(\lambda I - A)$$ 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.