If A and B commute so do A and B-1

Theorem 1.

Let A and B be commuting matricesMathworldPlanetmath. If B is invertiblePlanetmathPlanetmathPlanetmath, then A and B-1 commute, and if A and B are invertible, then A-1 and B-1 commute.


By assumptionPlanetmathPlanetmath


multiplying from the left and from the right by B-1 yields


The second claim follows similarly. ∎

The statement and proof of this result can obviously be extended to elements of any monoid. In particular, in the case of a group, we see that two elements commute if and only if their inversesPlanetmathPlanetmathPlanetmath do.

Title If A and B commute so do A and B-1
Canonical name IfAAndBCommuteSoDoAAndB1
Date of creation 2013-03-22 15:27:14
Last modified on 2013-03-22 15:27:14
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Theorem
Classification msc 15-00
Classification msc 15A27