Product distance matrix of a tree with matrix weights

Let T be a tree on n vertices and let the n−1 edges e1,e2,…,en−1 have weights that are s×s matrices W1,W2,…,Wn−1, respectively. For two vertices i, j, let the unique ordered path between i and j be pi,j=er1er2…erk. Define the distance between i and j as the s×s matrix Ei,j=∏p=1kWep. Consider the ns×ns matrix D whose (i,j)-th block is the matrix Ei,j. We give a formula for det⁡(D) and for its inverse, when it exists. These generalize known results for the product distance matrix when the weights are real numbers.