2.5. Inverse Matrices
81
2.5
Inverse Matrices
Suppose
A
is a square matrix. We look for an “
inverse matrix
”
A
1
of the same size, such
that
A
1
times
A
equals
I
. Whatever
A
does,
A
1
undoes. Their product is the identity
matrix—which does nothing to a vector, so
A
1
A
x
D
x
.
But
A
1
might not exist
.
What a matrix mostly does is to multiply a vector
x
. Multiplying
A
x
D
b
by
A
1
gives
A
1
A
x
D
A
1
b
.
This is
x
D
A
1
b
. The product
A
1
A
is like multiplying by
a number and then dividing by that number. A number has an inverse if it is not zero—
matrices are more complicated and more interesting. The matrix
A
1
is called “
A
inverse.”
DEFINITION
The matrix
A
is
invertible
if there exists a matrix
A
1
such that
A
1
A
D
I
and
AA
1
D
I:
(1)
Not all matrices have inverses
. This is the first question we ask about a square matrix:
Is
A
invertible? We don’t mean that we immediately calculate
A
1
. In most problems
we never compute it! Here are six “notes” about
A
1
.
Note 1
The inverse exists if and only if elimination produces
n
pivots
(row exchanges
are allowed). Elimination solves
A
x
D
b
without explicitly using the matrix
A
1
.
Note 2
The matrix
A
cannot have two different inverses. Suppose
BA
D
I
and also
AC
D
I
. Then
B
D
C
, according to this “proof by parentheses”:
B.AC/
D
.BA/C
gives
BI
D
IC
or
B
D
C:
(2)
This shows that a
leftinverse
B
(multiplying from the left) and a
rightinverse
C
(multi
plying
A
from the right to give
AC
D
I
) must be the
same matrix
.
Note 3
If
A
is invertible, the one and only solution to
A
x
D
b
is
x
D
A
1
b
:
Multiply
A
x
D
b
by
A
1
:
Then
x
D
A
1
A
x
D
A
1
b
:
Note 4
(Important)
Suppose there is a nonzero vector
x
such that
A
x
D
0
.
Then
A
cannot have an inverse.
No matrix can bring
0
back to
x
.
If
A
is invertible, then
A
x
D
0
can only have the zero solution
x
D
A
1
0
D
0
.
Note 5
A 2 by 2 matrix is invertible if and only if
ad
bc
is not zero:
2
by
2
Inverse:
a
b
c
d
1
D
1
ad
bc
d
b
c
a
:
(3)
This number
ad
bc
is the
determinant
of
A
. A matrix is invertible if its determinant is not
zero (Chapter 5). The test for
n
pivots is usually decided before the determinant appears.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
82
Chapter 2. Solving Linear Equations
Note 6
A diagonal matrix has an inverse provided no diagonal entries are zero:
If
A
D
2
6
4
d
1
:
:
:
d
n
3
7
5
then
A
1
D
2
6
4
1=d
1
:
:
:
1=d
n
3
7
5
:
Example 1
The 2 by 2 matrix
A
D
1 2
1 2
is not invertible. It fails the test in Note 5,
because
ad
bc
equals
2
2
D
0
. It fails the test in Note 3, because
A
x
D
0
when
x
D
.2;
1/
. It fails to have two pivots as required by Note 1.
Elimination turns the second row of this matrix
A
into a zero row.
The Inverse of a Product
AB
For two nonzero numbers
a
and
b
, the sum
a
C
b
might or might not be invertible. The
numbers
a
D
3
and
b
D
3
have inverses
1
3
and
1
3
. Their sum
a
C
b
D
0
has no inverse.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Neely

Click to edit the document details