  The inversion of extremely high order matrices has been a challenging task
because of the limited processing and memory capacity of conventional
computers. In a scenario in which the data does not fit in memory, it is worth
to consider exchanging less memory usage for more processing time in order to
enable the computation of the inverse which otherwise would be prohibitive. We
propose a new algorithm to compute the inverse of block partitioned matrices
with a reduced memory footprint. The algorithm works recursively to invert one
block of a $k \times k$ block matrix $M$, with $k \geq 2$, based on the
successive splitting of $M$. It computes one block of the inverse at a time, in
order to limit memory usage during the entire processing. Experimental results
show that, despite increasing computational complexity, matrices that otherwise
would exceed the memory-usage limit can be inverted using this technique.
