# Inverse matrix order of operation

If I have a rotation, $A$, and a translation, $B$, which I multiply like so

$C = AB$

Does the inverse of $C$ not only inverse the magnitudes of $A$ and $B$ but also become the reverse of operations?

e.g. $C^{-1} = BA$?

Does the inverse of a matrix also inverse the order of original operations (or am I conflating inverse with reverse)?

Yes. If you're compounding operations to make a matrix, then the inverse matrix will be the compound of the inverse operations, in the reverse order. So if $C = AB$ then $C^{-1} = B^{-1}A^{-1}$

Think of it geometrically. Taking a 2D example, if you have an object at the origin, and you want to move it +2 units in X then rotate around the origin by +45 degrees. To undo this transformation, you need to first rotate by -45 degrees then translate by -2 units in X.

• Ah! I was thinking the inverse preserved original multiplication order so it effectively created a mirror transformation, but it doesn't. Does this explain why it works to use it as the view matrix - because any subsequent rotations of the camera will end up being done first, thus keeping an object in view? Feb 25, 2018 at 21:14
• No, the inverse undoes the original transform, so multiplying the two together will give you an identity matrix. Not quite sure what you're asking about the view matrix, but a view matrix in general is the inverse of the camera's transform - it moves the camera to the origin with no rotation. Applying this matrix to the other objects in your scene means that as your camera moves through the world, after applying your view matrix the camera is a fixed point, and the world is moving around it with the inverse motion.
– russ
Feb 26, 2018 at 6:24