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otherwise the two systems of notation will be inconsistent. This implies that
Aa
+
Bc
AB
CD
ab
cd
Ab
+
Bd
=
·
(7.14)
Ca
+
Dc
Cb
+
Dd
which demonstrates how matrices must be multiplied. Here are the rules for
matrix multiplication:
Aa
+
Bc
A B
a
=
•
c
1
The top left-hand corner element
Aa
+
Bc
is the product of the top row of
the first matrix by the left column of the second matrix.
Ab
+
Bd
A B
b
=
•
d
2
The top right-hand element
Ab
+
Bd
is the product of the top row of the
first matrix by the right column of the second matrix.
a
=
•
Ca
+
Dc
c
C D
3
The bottom left-hand element
Ca
+
Dc
is the product of the bottom row
of the first matrix by the left column of the second matrix.
b
=
•
Cb
+
Dd
C D
d
4
The bottom right-hand element
Cb
+
Dd
is the product of the bottom row
of the first matrix by the right column of the second matrix.
It is now a trivial exercise to confirm Gauss's observation that T
1
×
T
2
=
T
2
×
T
1
, because if we reverse the transforms T
2
×
T
1
to T
1
×
T
2
we get
Aa
+
Bc
=
ab
cd
AB
CD
Ab
+
Bd
·
(7.15)
Ca
+
Dc
Cb
+
Dd
which shows conclusively that the product of two transforms is not co-
mmutative.
