We present a version of the matrix-tree theorem, which relates the
determinant of a matrix to sums of weights of arborescences of its directed
graph representation. Our treatment allows for non-zero column sums in the
parent matrix by adding a root vertex to the usually considered matrix directed
graph. We use our result to prove a version of the matrix-forest, or
all-minors, theorem, which relates minors of the matrix to forests of
arborescences of the matrix digraph. We then show that it is possible, when the
source and target vertices of an arc are not strongly connected, to move the
source of the arc in the matrix directed graph and leave the resulting matrix
determinant unchanged, as long as the source and target vertices are not
strongly connected after the move. This result enables graphical strategies for
factoring matrix determinants