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.

翻译：我们提出了矩阵-树定理的一个版本，该定理将矩阵的行列式与其有向图表示的枝干权之和联系起来。通过在通常考虑的矩阵有向图中添加一个根顶点，我们的处理方法允许母矩阵中存在非零列和。我们利用该结果证明了矩阵-森林定理（即全子式定理）的一个版本，该定理将矩阵的子式与矩阵有向图的枝干森林联系起来。接着，我们证明：当一条弧的源顶点和目标顶点并非强连通时，只要移动后源顶点和目标顶点仍非强连通，就可以在矩阵有向图中移动该弧的源顶点，且所得矩阵的行列式保持不变。这一结果为实现矩阵行列式分解的图形化策略提供了可能。