Showing both the analytical and numerical versions of a matrix

Question

I have found that the Defer function, among others such as HoldForm or Inactive, seems to be the key to something I want to do, but I have been unable to figure out. If I calculate the value of a variable and then print it using Defer, it correctly shows me the name of the variable. Doing so in a matrix however simply prints out the name of the matrix rather than the names of the variables within that matrix. I understand why this is the case, but I am unable to figure out how to apply this Defer to the elements of the matrix rather than to the matrix as a whole.

As an example:

a1 = 42;
mat = {{a1, a2}, {b1, b2}};
Print[Defer[MatrixForm[mat]]]
Print["***"]
Print[MatrixForm[mat]]

This currently prints:

mat
***
42  a2
b1  b2

What I would like is:

a1  a2
b1  b2
***
42  a2
b1  b2

This is a minimal example. In my case, I have several matrices, which are not necessarily 2x2, for which I would like to show both the analytical and numerical versions.

Answer 1

When you do

a1 = 42;
mat = {{a1, a2}, {b1, b2}};

Then mat has 42 already in there. So too late. I think you probably want

a1 = 42;
mat = HoldForm[MatrixForm@{{a1, a2}, {b1, b2}}];
Row[{mat, " = ", ReleaseHold@mat}]

It is important to note that after ReleaseHold@mat the MatrixForm wrapper is still there. So if you want to actually use mat for computation you need to do

mat = Flatten[List @@ ReleaseHold[mat], 1]

And now you can use mat in rest of code as it now has no wrapper and no Hold on it.

Update

Better ways than what I did (less typing) to remove MatrixForm and the Hold heads from mat is as suggested below by Somos. These are

mat = First@ReleaseHold[mat]

or

mat = Identity @@ ReleaseHold[mat]

Answer 2

Mathematica is a very rich symbolic language, and there are many ways to achieve the same result. However, given your description of the problem and my feeling that you are a novice in Mathematica, I would not immediately go to Defer or HoldForm, which are somehow more sophisticated and advanced functions, but would use a more common/idiomiatic approach with rules and ReplaceAll. This keeps the symbolic form of your matrix and its numerical values completely separated, combining them only when you explicitly want to do so.

mat = {{a1, a2}, {b1, b2}};
(* The following can come from some other function 
   that calculates values, e.g. vals = solveProblem[...]; *)
vals = {a1 -> 42, a2 -> 43, b2 -> -5};
Print[MatrixForm[mat]]
Print["***"]
Print[MatrixForm[mat /. vals]]