This question has been bothering me lately. I know the correct order to scale, rotate and transform a matrix but the question I have is very different. What if we first scale our rotation matrix by some value and then multiply it with our object? What will be its affect on our object?
The effect will be to rotate then scale, or scale then rotate your object, depending on what order you perform the multiplications.
Matrices (at least in less than 3 dimensions) carry super strong geometric intuitions. You can literally just think of them as holding transformations of space: be it translating, scaling, rotating, skewing or any combination of these. Multiplying two or more matrices, has the geometric equivalence of applying these transformations one after another. So if you multiple a rotation matrix * scaling matrix, you can think of it as scaling, then rotating space (or whatever vectors you put through this resulting matrix).
As to the name of the resulting matrix...
The definition of a rotation matrix is that it doesn't scale space nor reorder it's basis vectors [i,j,k]. More practically: rotation matrices don't change the magnitude of vectors or rearrange their [x,y,z] components.
This all means that if you scale a rotation matrix, then by definition it's no longer a "rotation" matrix --> now it's a matrix that rotates and scales.