The eigenvalues of a matrix $\mathbf{A}\in {\mathbb{C}}^{n\times n}$ are thus the $n$ roots of its characteristic polynomial $$P(\lambda) = det(\lambda\mathbf{I}-\mathbf{A}),$$ or $$P(\lambda)= \prod_{i=1}^{n}\left(\lambda_i-\lambda\right).$$ The set of these roots is called the spectrum and is denoted as $\lambda(\mathbf{A})$. If $\lambda(\mathbf{A})=\left\{\lambda_1,\lambda_2,\dots ,\lambda_n\right\}$ then we have $$det(\mathbf{A})= \lambda_1\lambda_2\dots\lambda_n,$$ and if we define the trace of $\mathbf{A}$ as $$Tr(\mathbf{A})=\sum_{i=1}^n a_{ii}$$ then $$Tr(\mathbf{A})=\lambda_1+\lambda_2+\dots+\lambda_n.$$