  We characterize complete intersection matrix Schubert varieties, generalizing
the classical result on one-sided ladder determinantal varieties. We also give
a new proof of the F-rationality of matrix Schubert varieties. Although it is
known that such varieties are F-regular (hence F-rational) by the global
F-regularity of Schubert varieties, our proof is of independent interest since
it does not require the Bott-Samelson resolution. As a consequence, this
provides an alternative proof of the classical fact that Schubert varieties in
flag varieties are normal and have rational singularities.
