Given a matrix and its characteristic polynomial, find its trace

Question

From Hoffman's Linear Algebra (section 6.3, exercise 9):

Let $A$ be a $n \times n$ matrix with characteristic polynomial $$f=(x-c_1)^{d_1} \cdots (x-c_k)^{d_k}.$$ Prove that $$\mbox{trace} (A) = c_1 d_1 + \cdots + c_k d_k$$

This is an exercise in the minimal polynomial section. Thus, in principle, it can be solved using methods covered in that section. Is there a "minimal polynomial"-method to solve this exercise?

Answer 1

No you cannot do this with minimal polynomials, because the minimal polynomial does not determine the trace. With respect to the characteristic polynomial mentioned in the question, the minimal polynomial may have lower exponents (but not $0$) instead of the $d_i$, but this is not reflected in the trace.