  Let $T$ be a tree with $n$ vertices. To each edge of $T$, we assign a weight
which is a positive definite matrix of some fixed order, say, $s$. Let $D_{ij}$
denote the sum of all the weights lying in the path connecting the vertices $i$
and $j$ of $T$. We now say that $D_{ij}$ is the distance between $i$ and $j$.
Define $D:=[D_{ij}]$, where $D_{ii}$ is the $s \times s$ null matrix and for $i
\neq j$, $D_{ij}$ is the distance between $i$ and $j$. Let $G$ be an arbitrary
connected weighted graph with $n$ vertices, where each weight is a positive
definite matrix of order $s$. If $i$ and $j$ are adjacent, then define
$L_{ij}:=-W_{ij}^{-1}$, where $W_{ij}$ is the weight of the edge $(i,j)$.
Define $L_{ii}:=\sum_{i \neq j,j=1}^{n}W_{ij}^{-1}$. The Laplacian of $G$ is
now the $ns \times ns$ block matrix $L:=[L_{ij}]$. In this paper, we first note
that $D^{-1}-L$ is always non-singular and then we prove that $D$ and its
perturbation $(D^{-1}-L)^{-1}$ have many interesting properties in common.
