Matrices, graphs and adjoints

Extract

Several writers have drawn attention to the fact that there is a connection between determinants and sets of loops in directed graphs. This fact is sometimes useful in evaluating the determinant of a matrix M, as was explained by Greenman in his recent article [1]. Towards the end of his article he introduces Δ_ij-subgraphs and relates them to the cofactors C_ij of the matrix M. Now if M is invertible, then M⁻¹ = adj M/det M, where the (j, i)-entry of the adjoint is just the cofactor C_ij. Hence one would expect to be able to give a graphical explanation for the fact that M(adj M/det M) = I, and this will be done in the present article.