  The right-quantum algebra was introduced recently by Garoufalidis, L\^e and
Zeilberger in their quantum generalization of the MacMahon master theorem. A
combinatorial proof of this identity due to Konvalinka and Pak, and also the
recent proof of the right-quantum Sylvester's determinant identity, make heavy
use of a bijection related to the first fundamental transformation on words
introduced by Foata. This paper makes explicit the connection between this
transformation and right-quantum linear algebra identities; applications
include a new combinatorial proof of the right-quantum matrix inverse theorem,
and two new results, the right-quantum Jacobi ratio theorem and a
generalization of the right-quantum MacMahon master thorem.
