# How do you prove that AB = BA if and only if AB is also symmetric?

##### 2 Answers

This statement is false.

#### Explanation:

Let:

#A=B=((0, 1, 0),(0,0,1),(1,0,0))#

Then:

#AB = BA = ((0, 1, 0),(0,0,1),(1,0,0))((0, 1, 0),(0,0,1),(1,0,0)) = ((0,0,1),(1,0,0),(0,1,0))#

which is not symmetric.

Let:

#A=B=((1,1),(0,1))#

Then:

#AB = BA = ((1,1),(0,1))((1,1),(0,1)) = ((1,2),(0,1))#

which is not symmetric.

See below

#### Explanation:

A corret proposition could be:

If

Suppose that

then

but

so

Note.

There are matrices