A Combinatorial Approach to Matrix Algebra

To most contemporary mathematicians matrices and linear transformations are practically interchangeable notions . Indeed, the mainstream 'Bourbakian' establishment, with its profound disdain of the concrete, goes as far as to frown at the mere mention of the word 'matrix'. To me, however, (as well as to a growing number of mathematical dissidents c~lled 'combinatorialists') a matrix has nothing whatsoever to do with that intimidating abstract concept called 'a linear transformation between linear vector spaces". Instead, an n x n matrix is the 'blueprint' of all the possible edges one can draw on n given vertices, a determinant is the 'weight' of all permutation graphs and matrix-products represent paths (details later). The purpose of this paper is to give a survey of this combinatorial interpretation of matrix algebra and_ to present elegant and illuminating proofs of five classical matrix identities . In 1965, Dominque Foata [4, 2] gave a beautiful combinatorial proof of the celebrated MacMahon master theorem, thus setting the stage for combinatorial matrix algebra. Recently, two other elegant proofs have appeared: Straubing's proof of Cayley-Hamilton [9], and Orlin [8], Garsia [6] and Temperley [10] independently found a combinatorial proof of the matrix tree theorem. I am going to present here new renditions of these three pearls, making them purely bijective and as succinct as possible. To them I am going to add two rubies of my own: a proof of det(AB) = (det A)(det B) and a new combinatorial proof (quite shorter than Foata's [5]) of Jacobi's det(eA) =eo-A.