  This paper studies the set of $n\times n$ matrices for which all row and
column sums equal zero. By representing these matrices in a lower dimensional
space, it is shown that this set is closed under addition and multiplication,
and furthermore is isomorphic to the set of arbitrary $(n-1)\times (n-1)$
matrices. The Moore-Penrose pseudoinverse corresponds with the true inverse,
(when it exists), in this lower dimension and an explicit representation of
this pseudoinverse in terms of the lower dimensional space is given. This
analysis is then extended to non-square matrices with all row or all column
sums equal to zero.
