  The dynamic matrix inverse problem is to maintain the inverse of a matrix
undergoing element and column updates. It is the main subroutine behind the
best algorithms for many dynamic problems whose complexity is not yet
well-understood, such as maintaining the largest eigenvalue, rank and
determinant of a matrix and maintaining reachability, distances, maximum
matching size, and $k$-paths/cycles in a graph. Understanding the complexity of
dynamic matrix inverse is a key to understand these problems.
  In this paper, we present (i) improved algorithms for dynamic matrix inverse
and their extensions to some incremental/look-ahead variants, and (ii) variants
of the Online Matrix-Vector conjecture [Henzinger et al. STOC'15] that, if
true, imply that these algorithms are tight. Our algorithms automatically lead
to faster dynamic algorithms for the aforementioned problems, some of which are
also tight under our conjectures, e.g. reachability and maximum matching size
(closing the gaps for these two problems was in fact asked by Abboud and V.
Williams [FOCS'14]). Prior best bounds for most of these problems date back to
more than a decade ago [Sankowski FOCS'04, COCOON'05, SODA'07; Kavitha
FSTTCS'08; Mucha and Sankowski Algorithmica'10; Bosek et al. FOCS'14].
  Our improvements stem mostly from the ability to use fast matrix
multiplication ``one more time'', to maintain a certain transformation matrix
which could be maintained only combinatorially previously (i.e. without fast
matrix multiplication). Oddly, unlike other dynamic problems where this
approach, once successful, could be repeated several times (``bootstrapping''),
our conjectures imply that this is not the case for dynamic matrix inverse and
some related problems. However, when a small additional ``look-ahead''
information is provided we can perform such repetition to drive the bounds down
further.
