A couple of ways to append the column col to mat.
In[1]:= mat = Partition[Range[12], 4]
Out[1]= {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}
In[2]:= col = {1, 1, 1};
In[3]:= Transpose[Join[Transpose[mat], {col}]]
Out[3]= {{1, 2, 3, 4, 1}, {5, 6, 7, 8, 1}, {9, 10, 11, 12, 1}}
In[4]:= Join[mat, List /@ col, 2]
Out[4]= {{1, 2, 3, 4, 1}, {5, 6, 7, 8, 1}, {9, 10, 11, 12, 1}}