Easy way to perform multiplication of two 2x2 matrices, that contain list elements?

Question

I have two 2x2 matrices, which elements are lists (so overall structure can be complicated). I need to perform matrix multiplication of these 2x2 matrices as if the elements didn't have its inner structure. But Mathematica's Dot[A, B] considers my matrices higher-dimentional and gives wrong result.

Then I introduced a function which literally perfoms 2x2 multiplication like this:

TwoMatrixProduct[a_, b_] := {
  {a[[1, 1]] b[[1, 1]] + a[[1, 2]] b[[2, 1]], 
   a[[1, 1]] b[[1, 2]] + a[[1, 2]] b[[2, 2]]},
  {a[[2, 1]] b[[1, 1]] + a[[2, 2]] b[[2, 1]], 
   a[[2, 1]] b[[1, 2]] + a[[2, 2]] b[[2, 2]]}}

and everything works fine, e.g.:

a = {a1, a2};
b = {b1, b2};
c = {c1, c2};
d = {d1, d2};
matrix2x2 = {{a,b},{c,d}}
TwoMatrixProduct[matrix2x2, matrix2x2]

(* {{{a1^2 + b1 c1, a2^2 + b2 c2}, 
     {a1 b1 + b1 d1, a2 b2 + b2 d2}}, 
    {{a1 c1 + c1 d1, a2 c2 + c2 d2}, 
     {b1 c1 + d1^2, b2 c2 + d2^2}}}  *)

Is there a better solution without manual typing of two matrices product?

Answer 1

matrix2x2
(* {{{a1, a2}, {b1, b2}}, {{c1, c2}, {d1, d2}}} *)
Map[Hold, matrix2x2, {2}]
(* {{Hold[{a1, a2}], Hold[{b1, b2}]}, {Hold[{c1, c2}], Hold[{d1, d2}]}} *)

And so:

ReleaseHold[Map[Hold, matrix2x2, {2}].Map[Hold, matrix2x2, {2}]]
(* {{{a1^2 + b1 c1, a2^2 + b2 c2},
     {a1 b1 + b1 d1, a2 b2 + b2 d2}},
    {{a1 c1 + c1 d1, a2 c2 + c2 d2},
     {b1 c1 + d1^2, b2 c2 + d2^2}}} *)