Calculate the determinant without matrix

Question

Sorry for my english. I have a question about how to calculate a determinant without the matrix representing the endomorphism in a certain basis

Example : If we have f : R2[X] --> R2[x]

P ---> f(P) = P+XP'

I want to calculate characteristic polynomial of f whitout any matrix if it's possible.

The formula is : p(c) = det(cId-f) where c is a real, and Id the identity application of R2[X]

I try to calculate cId-f, and I found cP-P-XP', is it good ? But then how to calculate det(cP-P-XP') without any matrix? Is it possible ?

Thanks a lot

Answer 1

Assuming by $R2[X]$ you mean polynomials in $X$ of degree $\le 2$, note that each $X^k$ is an eigenvector.