1.5 Inverses of Matrices, and Linear Systems
1
Chapter 1.
Vectors, Matrices, and Linear Spaces
1.5.
Inverses of Square Matrices
Definition 1.15.
An
n
×
n
matrix
A
is
invertible
if there exists an
n
×
n
matrix
C
such that
AC
=
CA
=
I
. If
A
is not invertible, it is
singular
.
Theorem 1.9. Uniqueness of an Inverse Matrix.
An invertible matrix has a unique inverse (which we denote
A
−
1
).
Proof.
Suppose
C
and
D
are both inverses of
A
. Then (
DA
)
C
=
IC
=
C
and
D
(
AC
) =
D
I
=
D
.
But (
DA
)
C
=
D
(
AC
) (associativity), so
C
=
D
.
QED
Example.
It is easy to invert an elementary matrix. For example, sup
pose
E
1
interchanges the first and third row and suppose
E
2
multiplies
row 2 by 7. Find the inverses of
E
1
and
E
2
.
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1.5 Inverses of Matrices, and Linear Systems
2
Theorem 1.10. Inverses of Products.
Let
A
and
B
be invertible
n
×
n
matrices. Then
AB
is invertible and
(
AB
)
−
1
=
B
−
1
A
−
1
.
Proof.
By associativity and the assumption that
A
−
1
and
B
−
1
exist, we
have:
(
AB
)(
B
−
1
A
−
1
) = [
A
(
BB
−
1
)]
A
−
1
= (
A
I
)
A
−
1
=
AA
−
1
=
I
.
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 Fall '04
 IgorDolgachev
 Linear Algebra, Vectors, Linear Systems, Matrices, Invertible matrix, Inverses

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