Using three different approaches, we analyze the complexity of various birational
maps constructed from simple operations (inversions) on square matrices of arbitrary size.
The first approach consists in the study of the images of lines, and relies mainly on
univariate polynomial algebra, the second approach is a singularity analysis, and the third
method is more numerical, using integer arithmetics. Each method has its own domain of
application, but they give corroborating results, and lead us to a conjecture on the
complexity of a class of maps constructed from matrix inversions.