abelian

Question

Every matrix is abelian under multiplication operation where all elements are real numbers

true or false

Answer 1

Before being Abelian , the given algebraic structure needs to be a group then we can talk about abelian group..

As far as multiplication is concerned ,

A . A^-1   =   I   where I is identity matrix ..

But  A^-1 exists provided A is an invertible matrix and hence det(A) should be non zero i.e. non singular matrix..Hence  A^-1  does not exist always , invalidating inverse property of group..

Hence it is not even group..Even for non singular matrix , it would not form abelian group as AB = BA need not be true always..

Comments

why |A| != 0 ??