  In this paper, we give novel certificates for triangular equivalence and rank
profiles. These certificates enable to verify the row or column rank profiles
or the whole rank profile matrix faster than recomputing them, with a
negligible overall overhead. We first provide quadratic time and space
non-interactive certificates saving the logarithmic factors of previously known
ones. Then we propose interactive certificates for the same problems whose
Monte Carlo verification complexity requires a small constant number of
matrix-vector multiplications, a linear space, and a linear number of extra
field operations. As an application we also give an interactive protocol,
certifying the determinant of dense matrices, faster than the best previously
known one.
