  In the early 1900's, Maillet proved that the image of any Liouville number
under a rational function with rational coefficients is again a Liouville
number. The analogous result for quadratic Liouville matrices in higher
dimension turns out to fail. In fact, using a result by Kleinbock and Margulis,
we show that among analytic matrix functions in dimension $n\ge 2$, Maillet's
invariance property is only true for M\"obius transformations with special
coefficients. This implies that the analogue in higher dimension of an open
question of Mahler on the existence of transcendental entire functions with
Maillet's property has a negative answer. On the other hand, extending a
topological argument of Erd\H{o}s, we prove that for any injective continuous
self mapping on the space of rectangular matrices, many Liouville matrices are
mapped to Liouville matrices. Dropping injectivity, we consider setups similar
to Alnia\c{c}ik and Saias and show that the situation depends on the matrix
dimensions $m,n$. Finally we discuss extensions of a related result by Burger
to quadratic matrices. We state several open problems along the way.
