a) Let A and B be symmetric matrices of the same size. Prove that AB is symmetric if and only AB=BA. b) Find symmetric 2 cdot 2 matrices A and B such that AB=BA.

Phoebe 2020-12-25 Answered
a) Let A and B be symmetric matrices of the same size.
Prove that AB is symmetric if and only AB=BA.
b) Find symmetric 22
matrices A and B such that AB=BA.
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Expert Answer

Layton
Answered 2020-12-26 Author has 89 answers
a) Suppose that AB is symmetric.
This means that (AB)T=AB
Since (AB)T=BTAT=BA
(because A and B are symmetric), we get AB=BA
as required

Suppose that Ab=BA
To prove that AB is symmetric, we will prove that AB=(AB)T.
Since (AB)T=BTAT=BA=AB
where we used that A and B are symmetric in the second equality and the assumption Ba=Ab
in the last equation and the assumption BA=BA in the last equality,
we get that (AB)T=AB
as required
b) Let
A=[1110] and B=[0001]
Then AB=[1110][0001]=[0100]
and AB=[0001][1110]=[0010]
so ABBA
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