Math 20F Linear Algebra
Lecture 8: 2.2 The inverse of a matrix.
Definition
: The identity matrix is
I
= [
δ
ij
], where
δ
ij
= 1 if
i
=
j
and
δ
ij
= 0 if
i
6
=
j
:
I
=
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
,
in case 4
×
4
.
Definition
: The transpose
A
T
is the matrix with rows and columns interchanged,
(
A
T
)
ij
=(
A
)
ji
Example:
If
A
=
1
2
3

2
0

1
4
5
2
then
A
T
=
1

2
4
2
0
5
3

1
2
.
Which is true?
(
AB
)
T
=
A
T
B
T
or (
AB
)
T
=
B
T
A
T
?
An
n
×
n
matrix
A
is said to be
invertible
if there is an
n
×
n
matrix
A

1
such
that
(2.2.1)
A

1
A
=
AA

1
=
I
where
I
is the identity matrix. The matrix
A

1
called the
inverse
of
A
is unique.
In fact if
BA
=
I
then
B
=
BI
=
B
(
AA

1
) = (
BA
)
A

1
=
IA

1
=
A

1
.
Not all
n
×
n
matrices are invertible.
A matrix which is not invertible is called
singular
. An invertible matrix is called
nonsingular
.
An
inverse
of a transformation
x
→
T
(
x
) is a transformation which takes you
back
T
(
x
)
→
x
.
The condition
A

1
A
=
I
says that the inverse of the linear
transformation
x
→
A
x
is the linear transformation
y
→
A

1
y
.
In fact, if we
compose
x
→
A
x
with
y
→
A

1
y
we get
x
→
A
x
→
A

1
(
A
x
)=(
AA

1
)
x
=
I
x
=
x
.
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 Spring '03
 BUSS
 Linear Algebra, Algebra, Invertible matrix, A1, n×n matrix A1

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