  The \emph{distance matrix} of a simple connected graph $G$ is
$D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices $i$ and
$j$ in $G$. We consider a weighted tree $T$ on $n$ vertices with edge weights
are square matrix of same size. The distance $d_{ij}$ between the vertices $i$
and $j$ is the sum of the weight matrices of the edges in the unique path from
$i$ to $j$. In this article we establish a characterization for the trees in
terms of rank of (matrix) weighted Laplacian matrix associated with it. Then we
establish a necessary and sufficient condition for the distance matrix $D$,
with matrix weights, to be invertible and the formula for the inverse of $D$,
if it exists. Also we study some of the properties of the distance matrices of
matrix weighted trees in connection with the Laplacian matrices, g-inverses and
eigenvalues.
