Theorem

Let $\mathbf{A}=\{ a_{ij} \}$ be a $k\times k$ matrix of real elements with eigenvalues $\lambda_{1},\dots,\lambda_{k}$ (counting multiplicities) then $$\mathrm{Tr}(\mathbf{A})=\sum_{i=1}^{k}a_{ii}=\sum_{i=1}^{k}\lambda_{i}$$and $$\det(\mathbf{A})=\prod_{i=1}^{k}\lambda_{i}$$