# Why are rotational matrices not commutative?

Is there any intuition why rotational matrices are not commutative? I assume the final rotation is the combination of all rotations. Then how does it matter in which order the rotations are applied?

Here is a picture of a die:

Now let’s spin it $90^\circ$ clockwise. The die now shows

After that, if we flip the left face up, the die lands at

Now, let’s do it the other way around: We start with the die in the same position:

Flip the left face up:

and then $90^\circ$ clockwise

If we do it one way, we end up with $3$ on the top and $5, 6$ facing us, while if we do it the other way we end up with $2$ on the top and $1, 3$ facing us. This demonstrates that the two rotations do not commute.

Since so many in the comments have come to the conclusion that this is not a complete answer, here are a few more thoughts:

• Note what happens to the top number of the die: In the first case we change what number is on the left face, then flip the new left face to the top. In the second case we first flip the old left face to the top, and then change what is on the left face. This makes two different numbers face up.
• As leftaroundabout said in a comment to the question itself, rotations not commuting is not really anything noteworthy. The fact that they do commute in two dimensions is notable, but asking why they do not commute in general is not very fruitful apart from a concrete demonstration.