Theorem (Trace and determinant in terms of eigenvalues)

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

Theorem (Distinct eigenvalues have linearly independent eigenvectors)

For $T\in\mathscr{L}(V)$, let $\lambda_1,\cdots,\lambda_k\in\mathbb{F}$ be distinct eigenvalues of $T$ with corresponding eigenvectors $v_1,\cdots,v_k$. The vectors $v_1,\cdots,v_k$ are linearly independent.

Theorem (Operators on $\mathbb{C}$-vector spaces have an eigenvalue)

If $V$ is a Finite Dimensional $\mathbb{C}$-Vector Space and $T\in\mathscr{L}(V)$, then T has an eigenvalue or $$\lambda\text{ is an eigenvalue of }T\iff\exists v\in V, \ v\not=0, \mbox{ s.t. }(T-\lambda I_V)(v)=0$$