Raise Warning/Error on invalid Matrix Addition and Multiplication

Question

Consider the following code:

DiagonalMatrix[{1, 1, 1}] - 1

Mathematica interprets it in the way, that it substracts 1 from every matrix field while mathematically this operation is not defined. Is it possible to raise an error or at least a warning?

Furthermore

invalid = DiagonalMatrix[{1, 1, 1}].a
invalid /. a -> 4

produces an invalid matrix multiplication. I also want to raise an error or warning if this happens.

Answer 1

That would mean to remove the Listable attribute from Plus. You could do it as follows, but I would not recommend that.

Unprotect[Plus];
ClearAttributes[Plus, Listable];
Plus[_?MatrixQ, __?(! MatrixQ[#] &)] := $Failed

Now, you have

a = IdentityMatrix[3];
Plus[a, a]
Plus[a, 1]

{{2, 0, 0}, {0, 2, 0}, {0, 0, 2}}

$Failed

Really, you should embrace the feature of listability rather than risking to break the whole system.

A better option might be to use the symbol CirclePlus instead; it has no built-in meaning. (You have to restart the kernel first in order to reset Plus to its original definition.)

CirclePlus::errorname = "Error message.";

(* This ensures that `CirclePlus` has in general the same functionality as `Plus`. *)
CirclePlus[x___] := Plus[x]; 

(* This this adds a special rule for undesired additions. *)
CirclePlus[a_?MatrixQ, b__?(! MatrixQ[#] &)] := (
 Message[CirclePlus::errorname]; 
 Plus[a, b]
 ) 

Now you have

a = IdentityMatrix[3];
a⊕a
a⊕1

{{2, 0, 0}, {0, 2, 0}, {0, 0, 2}}

{{2, 1, 1}, {1, 2, 1}, {1, 1, 2}}