In[5]:= m = {{0.768576, -2.62804*10^-11}, {2.44999*10^8, 0.369943}}
Out[5]= {{0.768576, -2.62804*10^-11}, {2.44999*10^8, 0.369943}}
In[6]:= mr = Rationalize[m, 10^-16]
Out[6]= {{12009/15625, -(1/38051171215)}, {244999000, 369943/1000000}}
In[7]:= mrinv = Inverse[mr]
Out[7]= {{43989888852471078125/34575194689676811341, 3125000000/
34575194689676811341}, {-(29132809051574328125000000000/
34575194689676811341), 91391303024187000000/34575194689676811341}}
In[8]:= minv = N @ mrinv
Out[8]= {{1.2723, 9.03827*10^-11}, {-8.42593*10^8, 2.64326}}
In[9]:= m.minv
Out[9]= {{1., 9.12871*10^-23}, {0., 1.}}