Why are there two different forms of the Characteristic Polynomial?

Question

Excuse me for my ignorance towards this topic. Unfortunately my professor waited to the last of day class to cover this important chapter.

He merely told us that the Characteristic Polynomial equation is:

(1) $$p(\Lambda)=det(A-\Lambda_i I_n)$$

Yet, my textbook claims this Characteristic Polynomial equation to be:

(2) $$p(\Lambda)=det(\Lambda_i I_n - A)$$

Where $n$ is an $n\times n$ matrix. In doing practice problems I've found the two the two equations to be equivalent. Unfortunately my textbook only provides the proof for the second equation. Therefore my question is: is one technically more "correct" or "formal" than the other? And why then are there two permutations of such a basic forumla--even if they are equivalent?

Answer 1

To repeat what Wikipedia says about characteristic polynomials:

We consider an $n×n$ matrix $A$. The characteristic polynomial of $A$, denoted by $p_A(t)$, is the polynomial defined by

$${\displaystyle p_{A}(t)=\det \left(t{{I}}-A\right)}\tag{1}$$
where $I$ denotes the $n\times n$ identity matrix.

Some authors define the characteristic polynomial to be $$\det(A - t I)\tag{2}$$ That polynomial differs from the one defined here by a sign $(−1)^n$, so it makes no difference for properties like having as roots the eigenvalues of $A$; however the current definition always gives a monic polynomial, whereas the alternative definition always has constant term $\det(A)$.

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I would say that one usually uses (1) for getting a monic polynomial. (2) is probably "more natural" for some people since when looking for an eigenvecotr, one solves $Av=\lambda v$ which is $(A-\lambda I)v=0$. To my knowledge, there is no anything deep among these two alternatives.