A combinatorial proof of the trace Cayley-Hamilton theorem

The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed graph D(A), where the algebraic behavior of A is reflected in the combinatorial properties of D(A). In particular, the determinant and characteristic polynomial of A admit elegant formulations in terms of sign-weighted sums over linear subdigraphs of D(A), thereby providing a graphical interpretation of fundamental algebraic quantities. Building upon this correspondence, we establish a combinatorial proof of the trace Cayley-Hamilton theorem. This theorem furnishes explicit trace identities linking the coefficients of the characteristic polynomial of A with the traces of its successive powers.