Question

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

Answer 1

This statement is false.

Explanation:

`bb(3 xx 3)` matrices

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.

`color(white)()`
`bb(2 xx 2)` matrices

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.

Answer 2

See below

Explanation:

A corret proposition could be:

If `A` is symmetric `AB=BA iff B` is symmetric

Suppose that `A,B` are non null matrices and `AB = BA` and `A` is symmetric but `B` is not

then

`AB = (AB)^T = B^TA^T = B A`

but `A = A^T`

so

`B^TA^T-BA=0->(B^T-B)A=0->B^T=B` which is an absurd. So `B` must be also symmetric.

Note.

There are matrices `A,B` not symmetric such that verify

`AB=BA`. Example

`A =((4, -1),(1/2, 3))`
`B = ((1, 2),(-1, 3))`

`AB=BA =((5, 5),(-5/2, 10))`