Let me see if I understand. The result of

list = {#, # + 1, "constantmatrix"} & /@ Range[5]

must be

{{1, 2, "constantmatrix"}, {2, 3, "constantmatrix"}, {3, 4, 
  "constantmatrix"}, {4, 5, "constantmatrix"}, {5, 6, 
  "constantmatrix"}}

and the final result of

Apply[f, list, 1]

must be

{48, 50, 52, 54, 56}

It would be much better for us if you could explain why you need that string in the intermediate result. Is this only a challenge?

Still struggling with this Frank! If you look at the code below:

 In[30]:= Clear[constantmatrix]
    list = {#, # + 1, constantmatrix} & /@ Range[100];
    ByteCount[list]
    constantmatrix = RandomReal[1000, {100, 100}];
    ByteCount[constantmatrix]
    ByteCount[list]

Out[32]= 10496

Out[34]= 80152

Out[35]= 8025696

Here list blows up in size as soon as I define constant matrix - because the same matrix is repeated in list. However knowing that this is the same list, is there an efficient way to define the function (or some other way) - which utilizes the information that constantmatrix will be the same for every evaluation of the function:

f[a_, b_, constant_] := a + b + Plus @@ Flatten[constant]
Apply[f, list, 1]

Sorry if I am being unclear.