An
n
n
-fir is an associative ring in which every
n
n
-generator right ideal is free of unique rank. Matrix localization of a ring involves the adjunction of universal inverses to certain matrices over the ring, so that a new ring results over which the matrices have inverses, but so that the minimum of additional relations is imposed. A full matrix is a square matrix which, when considered as an endomorphism of a free module, cannot be factored through a free module of smaller rank. The main result of this paper is that if the original ring is an
n
n
-fir with
n
>
2
k
n > 2k
and if we form a matrix localization by adjoining universal inverses to all full matrices of size
k
k
, then the resulting ring is an
(
n
−
2
k
)
(n - 2k)
-fir. This generalizes an announced result of V. N. Gerasimov. There are related results on the structure of the universal skew field of fractions of a semifir.