A combinatorial approach to matrix algebra (Q1086566)

In this paper, the author takes the combinatorial point of view that ''an \(n\times n\) matrix is the 'blueprint' of all possible edges one can draw on n given vertices, a determinant is the 'weight' of all permutation graphs and matrix-products represent paths''. After surveying such a combinatorial interpretation of matrix algebra, he presents new combinatorial proofs to five classical matrix identifies; namely, (1) MacMahon's master theorem, (2) the Cayley-Hamilton theorem, (3) the matrix tree theorem, (4) \(\det (AB)=(\det A)/\det B)\) and (5) \(\det (e^ A)=e^{tr(A)}\).