# What happens if first we scale a rotation matrix and then multiply it with an object?

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?

• This looks like a question you could answer by trying it and observing the results. What do you observe when you try this? Jun 22 at 20:04
• @DMGregory is there a tool (desktop or online) that allows playing around and observing this stuff? I would be interested in experimenting without having to setup complicated C libraries and opengl windows. Jun 22 at 20:06
• Like literally any game engine? If you're developing a game, you should already have this at your fingertips. If you're not developing a game, then you might be on the wrong website. Jun 22 at 20:08
• I actually use this place to learn about linear algebra, 3D math etc. from a practical aspect instead of the more theoretic approach you'd find in the math stackexchange! Jun 22 at 20:13
• I'm not a big math person. But I'm pretty sure the definition of a rotation matrix is that it doesn't scale [i,j,k]. Or at least practically: vectors that are transformed by it don't change magnitude and the coordinate system doesn't change order. So 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. Jun 22 at 20:38

##### Effect

The effect will be to rotate then scale, or scale then rotate your object, depending on what order you perform the multiplications.

###### Explanation

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.