Theorem: Let A and B be matrices. (AB)T =
B T AT .
Proof:
First observe that the ij entry of AB can be written as
n
X
(AB)ij =
aik bkj .
k=1
Furthermore, if we transpose a matrix we switch
the rows and the columns. These facts together
mean that we can write
(AB)T
ij
= (AB)ji =
n
X
ajk bki
k=1
and
(B T AT )ij =
n
X
(B T )ik (AT )kj =
k=1
n
X
bki ajk .
k=1
From here is is clear that the ij entry of the left
and right sides are equal. Therefore the matrices
are equal.
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