Element assignment in matrix function

Question

I have to define a function of two variables that yields a high dimensional matrix, with each entry containing a scalar function, that uses its coordinates in the matrix as input. like this:

f[x_,a_] := {{g[1,1][x]*a, g[1,2][x]*a}, {g[2,1][x]*a, g[2,2][x]*a}}

Is there any way I can assign the elements in a smart way without mapping the whole matrix by hand?

EDIT: I've been asked for an example of g. This is taken out of the original code:

g[n_, m_] := Table[Chop[
    I*Conjugate[h[[n]][[i]]].(hx[[m]][[i]] - h[[m]][[i]])/
      step, 3*10^-3],
   {i, 1, Length[qvals]}];

Answer 1

gs = Flatten @ Table[With[{m = m, n = n}, Sin[m #/n^2] &], {m, 2}, {n, 2}];

{Sin[#1/1^2] & , Sin[#1/2^2] & , Sin[(2*#1)/1^2] & , Sin[(2*#1)/2^2] &}

f[x_, a_] = a Through[gs[x]];
f[u, b]

{b*Sin[u], b*Sin[u/4], b*Sin[2*u], b*Sin[u/2]}