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

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Answer 1 · 2016-07-29T05:56:38

This statement is false.

#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 · 2016-07-29T08:12:30

See below

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))#