Math 313 Lecture #4
§
1.3: Matrix Arithmetic, Part II
§
1.4: Matrix Algebra, Part I
More about Matrix Multiplication.
Let
A
= (
a
ij
) be an
m
×
n
matrix and
B
= (
b
ij
)
an
n
×
r
matrix. The (
i, j
) entry of the product
AB
is a scalar product:
n
k
=1
a
ik
b
kj
=
a
(
i,
:)
b
j
.
That is, as the index
k
varies from 1 to
n
, the values
a
ik
are in the
i
th
row of
A
, and the
values
b
kj
are in the
j
th
column of
B
.
This means that the (
i, j
) entry of
AB
is the scalar product of the
i
th
row of
A
with the
j
th
column of
B
(as shown above).
Now as the (
i, j
) entry of
AB
is of the form
a
(
i,
:)
b
j
, then the
j
th
column of
AB
is the
column vector
Ab
j
=
a
(1
,
:)
b
j
a
(2
,
:)
b
j
.
.
.
a
(
m,
:)
b
j
=
a
11
a
12
. . .
a
n
1
a
21
a
22
. . .
a
2
n
.
.
.
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
. . .
a
mn
b
1
j
b
2
j
.
.
.
b
nj
.
Thus the columns of the product of
A
with
B
are
AB
= (
Ab
1
, Ab
2
, . . . , Ab
n
)
.
The Rules of Matrix Algebra.
Let
α
and
β
be scalars, and
A
,
B
, and
C
matrices
for which the indicated operations are defined. Then
1.
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 Fall '10
 na
 Linear Algebra, Algebra, Multiplication, Scalar, Ring

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