The trace Cayley-Hamilton theorem

In this expository paper, various properties of matrix traces, determinants and adjugate matrices are proved, including the trace Cayley-Hamilton theorem, which says that $$k{c}_k+\sum _{i=1}^k\mathrm{Tr}(A^i)c_{k-i}=0\text{for every }k\in \mathbb{N}$$ whenever $A$ is an $n\times n$-matrix with characteristic polynomial $\det (t{I}_n-A)={\sum }_{i=0}^n{c}_{n-i}{t}^i$ over a commutative ring $\mathbb{K}$. While the results are not new, some of the proofs are. The proofs illustrate some general techniques in linear algebra over commutative rings.