Let A and B be
n ⨯ n
orthogonal matrices. Show that both AB and BA are also orthogonal.
We want to show that
AB
and
(AB)T
are . Indeed, we have the following.
AB(AB)T = AB
= A
AT = =
We conclude
AB
is orthogonal.
Similarly, we want to show that
BA
and
(BA)T
are . Again, we have the following.
BA(BA)T = BA
= B