  Asymptotically tight lower bounds are derived for the I/O complexity of a
general class of hybrid algorithms computing the product of $n \times n$ square
matrices combining ``\emph{Strassen-like}'' fast matrix multiplication approach
with computational complexity $\Theta{n^{\log_2 7}}$, and ``\emph{standard}''
matrix multiplication algorithms with computational complexity
$\Omega\left(n^3\right)$. We present a novel and tight
$\Omega\left(\left(\frac{n}{\max\{\sqrt{M},n_0\}}\right)^{\log_2
7}\left(\max\{1,\frac{n_0}{M}\}\right)^3M\right)$ lower bound for the I/O
complexity a class of ``\emph{uniform, non-stationary}'' hybrid algorithms when
executed in a two-level storage hierarchy with $M$ words of fast memory, where
$n_0$ denotes the threshold size of sub-problems which are computed using
standard algorithms with algebraic complexity $\Omega\left(n^3\right)$.
  The lower bound is actually derived for the more general class of
``\emph{non-uniform, non-stationary}'' hybrid algorithms which allow recursive
calls to have a different structure, even when they refer to the multiplication
of matrices of the same size and in the same recursive level, although the
quantitative expressions become more involved. Our results are the first I/O
lower bounds for these classes of hybrid algorithms. All presented lower bounds
apply even if the recomputation of partial results is allowed and are
asymptotically tight.
  The proof technique combines the analysis of the Grigoriev's flow of the
matrix multiplication function, combinatorial properties of the encoding
functions used by fast Strassen-like algorithms, and an application of the
Loomis-Whitney geometric theorem for the analysis of standard matrix
multiplication algorithms.
  Extensions of the lower bounds for a parallel model with $P$ processors are
also discussed.
