Two by two matrices are an abelian group under addition. But although multiplication is associative and distributive, and the identity matrix does the job of 1, multiplication isn't commutative:

Example:

Also, there are non-zero matrices which don't have inverses, so that axiom fails too.

Analysis WebNotes by John Lindsay Orr.
Comments to the author: jorr@math.unl.edu

All contents copyright (C) 1996 John L. Orr
University of Nebraska--Lincoln
All rights reserved

Last modified: May 1996