  It is known that the Cabibbo-Kobayashi-Maskawa (CKM) $n\times n$ matrix can
be represented by a real matrix iff there is no CP-violation, and then the
Jarlskog invariants vanish. We investigate sufficient conditions for the
opposite statement to hold, paying particular attention to degenerate cases. We
find that higher Jarlskog invariants are needed for $n\geq 4$. One generic
sufficient condition is provided by the existence of a so-called echelon cross.
