The property that det(A) = prod of A's eigenvalues, and tr(A) = sum of A's eigenvalues

Question

Do these two properties fail to be true, if A's characteristic polynomial fails to split?

If so, then do we usually work in a vector space with the ground field = $\mathbb{C}$, when we want to use these two equations?

Thanks,

Answer 1

Given an irreducible degree $2$ monic polynomial over the reals it is possible to find a $2\times 2$ matrix having the given polynomial as its characteristic polynomial.

Take $x^2-2x+2$ as the polynomial. A matrix having it as its characteristic polynomial is the “companion matrix” $$ \begin{bmatrix} 0 & -2 \\ 1 & 2 \end{bmatrix} $$ The matrix has no real eigenvalue, so the sum of the real eigenvalues is $0$ and the product is $1$ (this is the convention for the empty sum and product); however the trace is $2$ and the determinant is $2$.

If the characteristic polynomial splits completely, then the statement is true.