How to make the determinant formula visible even in cases when det=0

Question

When I compute the determinant analytically in Mathematica, I do it with

Det[{{a, b}, {c, d}}]

which gives the output

-b c + a d

I would like to see this formula even if the determinant is zero as

Det[{{a, b}, {-a, -b}}]

which yields the output

0

instead of something like

-b (-a) + a (-b)

This is just a small example, but my matrices are a little larger, and I need to figure out analytically why some determinants are zero and some are not.

Any ideas?

Answer 1

One possibility is to temporarily inactivate the arithmetic operators, like so:

Block[{Times = Inactive[Times], Plus = Inactive[Plus]}, 
      Det[{{a, b}, {-a, -b}}]]
   a*(-1*b)+-1*b*(-1*a)

Answer 2

You could try:

m = {{a, b}, {-a, -b}};

Det[Map[ToBoxes, m, {2}]] // DisplayForm
(* -b (-a)+a (-b) *)