Solving for N in your system is very easy: it only appears once, in
the first equation.
If we call your two equations eq1 and eq2, then solve(eq1,N) will do
it (assuming that n is distinct from N, just as t and T are distinct
in your example).
On the other hand, solving for F symbolically seems intractable in
general, since there are subexpressions of the form F*exp(F*k) and
F+exp(F), making the equations non-algebraic (and not algebraic in
exp(F), either). But perhaps there is some structure in these
particular equations that I am missing. Or perhaps they can be solved
in terms of the Lambert W-function, which I suspect Maple supports
particularly well.
By the way, instead of using T:matrix([...]) and T[1,n], you can use
T:[...], T[n] -- these are called lists in Maxima.
-s