1.3/1.5 MATRIX ARITHMETIC/ELEMENTARY MATRICES
KIAM HEONG KWA
p. 27 The scalar
a
ij
of a matrix
A
=
(
a
ij
)
=
a
11
a
12
· · ·
a
1
n
a
21
a
22
· · ·
a
2
n
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
· · ·
a
mn
is called the
(
i
,
j
)
entry
of
A
.
p. 29 Two
m
×
n
matrices
A
=
(
a
ij
)
and
B
=
(
b
ij
)
are said to be
equal
if and only if their corresponding entries agree:
A
=
B
iff
a
ij
=
b
ij
∀
i, j.
p. 29 (Scalar multiplication) For any matrix
A
=
(
a
ij
)
and any scalar
α
,
αA
=
(
αa
ij
)
.
pp. 2930 (Addition) The
sum
of two
m
×
n
matrices
A
=
(
a
ij
)
and
B
=
(
b
ij
)
is the matrix
A
+
B
=
(
a
ij
+
b
ij
)
.
The
difference
of
A
and
B
is the matrix
A
+ (

1)
B
.
The
m
×
n
matrix
0
whose entries are all zero is called the
zero
matrix
and it acts as an additive identity:
A
+ 0 = 0 +
A
=
A.
The matrix

A
= (

1)
A
is the
additive inverse
of
A
:
A
+ (

A
) = (

A
) +
A
= 0
.
pp. 3839 The
transpose
of an
m
×
n
matrix
A
=
(
a
ij
)
is the
n
×
m
matrix
A
T
=
(
b
ij
)
whose
(
i, j
)
entry is given by
b
ij
=
a
ji
. In particular,
the transpose of a row vector is a column vector. It follows that a
row vector
~
a
i
is the
i
th row of
A
if and only if
~
a
T
i
is the
i
th column
of
A
T
. Also,
(
A
T
)
T
=
A.
A square matrix
A
is said to be
symmetric
if and only if
A
T
=
A
.
Date
: June 18, 2011.
1
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KIAM HEONG KWA
p. 35 (Matrix multiplication) Let
A
=
(
a
ij
)
be an
m
×
n
matrix and
B
=
(
b
ij
)
be an
n
×
r
matrix. The
product
of
A
and
B
is the
m
×
r
matrix
AB
=
C
=
(
c
ij
)
whose
(
i, j
)
entry is
c
ij
=
n
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 Summer '08
 KIM
 Linear Algebra, Algebra, Matrices, Scalar, cij, eij

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