  The row (resp. column) rank profile of a matrix describes the staircase shape
of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a
recursive Gaussian elimination that can compute simultaneously the row and
column rank profiles of a matrix as well as those of all of its leading
sub-matrices, in the same time as state of the art Gaussian elimination
algorithms. Here we first study the conditions making a Gaus-sian elimination
algorithm reveal this information. Therefore, we propose the definition of a
new matrix invariant, the rank profile matrix, summarizing all information on
the row and column rank profiles of all the leading sub-matrices. We also
explore the conditions for a Gaussian elimination algorithm to compute all or
part of this invariant, through the corresponding PLUQ decomposition. As a
consequence, we show that the classical iterative CUP decomposition algorithm
can actually be adapted to compute the rank profile matrix. Used, in a Crout
variant, as a base-case to our ISSAC'13 implementation, it delivers a
significant improvement in efficiency. Second, the row (resp. column) echelon
form of a matrix are usually computed via different dedicated triangular
decompositions. We show here that, from some PLUQ decompositions, it is
possible to recover the row and column echelon forms of a matrix and of any of
its leading sub-matrices thanks to an elementary post-processing algorithm.
