Unit II: Matrix Algebra
1. Algebraic Operations with Matrices
addition of matrices, multiplication of a matrix by a scalar
Definition of product of matrices: If
A
is an
m
×
n
matrix, and
B
is an
n
×
p
matrix
with columns
b
1
, . . . ,
b
p
∈
R
n
, then
AB
is the
m
×
p
matrix with columns
A
b
1
, . . . , A
b
p
.
In other words,
AB
=
A
[
b
1
, . . . ,
b
p
] = [
A
b
1
, . . . , A
b
p
]
.
Lemma.
If
A
is an
m
×
n
matrix, and
B
is an
n
×
p
matrix, then
(
AB
)
ik
=
n
X
j
=1
A
ij
B
jk
,
1
≤
i
≤
m,
1
≤
k
≤
p.
Proofsketch.
Use the definition of
A
x
with
x
replaced by the
k
th column of
B
.
Rules for operations on matrices: matrix addition is associative and commutative, ma
trix multiplication is associative, and the usual distributive rules hold.
However, matrix
multiplication is NOT commutative.
vectors as matrices, row vectors, column vectors, vector form of a system of linear equa
tions,
A
x
=
0
,
A
x
=
b
identity matrix, diagonal matrix, upper triangular matrix, lower triangular matrix
transpose
A
t
of a matrix
A
, transpose of a product of matrices, (
AB
)
t
=
B
t
A
t
, symmetric
matrix (
A
t
=
A
)
2. Invertible Matrices
An
n
×
n
matrix
A
is
invertible
if there is a matrix
B
such that
AB
=
BA
=
I
. The
matrix
B
is called the
inverse
of
A
and denoted by
A

1
. The next lemma shows that the
inverse
B
of
A
is unique, so that
A

1
is well defined.
Lemma.
If
BA
=
I
, and
AC
=
I
, then
B
=
C
.
Proof.
B
=
BI
=
B
(
AC
) = (
BA
)
C
=
IC
=
C
.
Lemma.
If the square matrix
A
is invertible, then
A

1
is invertible, and
(
A

1
)

1
=
A
.
Proof.
This follows from
AA

1
=
I
=
A

1
A
.
Lemma.
If the
n
×
n
matrices
A
and
B
are invertible, then
AB
is invertible, and
(
AB
)

1
=
B

1
A

1
.
Lemma.
An
n
×
n
matrix
A
is invertible if and only if its transpose
A
t
is invertible, and
(
A
t
)

1
= (
A

1
)
t
.
Proofsketch
. Use (
AB
)
t
=
B
t
A
t
and (
AB
)

1
=
B

1
A

1
.
If
A
is invertible, then the system
A
x
=
b
has a unique solution, namely,
x
=
A

1
b
.
This property of having unique solutions characterizes invertible matrices.
1