If a matrix $A$ is Diagonalizable, then $\exists$ a Non singular matrix $P$ such that

$$D=P^{-1}AP$$ Now taking Trace on both sides

$$Tr(D)=Tr(P^{-1}AP)=Tr(APP^{-1})=Tr(A)$$

Now since $D$ is Diagonal matrix with diagonal elements as eigen values we have

$Tr(A)$ as Sum of eigen values of $A$.

But how to prove this if $A$ is not diagonalizable?