What more do we need from our matrix class? We have addition,
subtraction and multiplication, and thus composition of generic
freemodulemaps. We're going to want to solve linear equations with
or without fields underneath, so we're going to want an implementation
of the Gaussian algorithm as well as most probably Smith normal
form. Determinants are cool, and these are to be calculated either
with the Gaussian algorithm or some other goodish method.
We'll want generic linear equation solving, returning one solution,
any solution really, or nothing. Basically, this is asking for the
preimage of a given vector over the given map, so
a_11 x_1 + .. + a_1n x_n = y_1
...
a_m1 x_1 + .. + a_mn a_n = y_m
has really x_1,...,x_n as a preimage of the vector y_1,..,y_m under
the map (a_ij), since obviously y_1,..,y_m = (a_ij) x_1,..,x_n
So, generic linear equation solving boils down to the function
