12.3
Inverse Matrices
Two matrices A and A−1 are called inverses if
AA−1 I
A−1 A I
and
where I denotes the identity matrix of the appropriate size.
For example, the matrices
"
A
3
2
7
5
#
A
and
−1
"
5
−2
1
0
0
1
#
1
0
0
1
#
−7
3
#
are inverses, since
"
If we think of the identity matrix I as
being analogous to the number 1, then
inverses are analogous to a reciprocal pair
of numbers, such as 2/3 and 3/2.
AA−1 and
"
A A −1
3
2
5
−2
7
5
#"
−7
3
5
−2
#"
3
2
−7
3
#
7
5
#
"
"
I
I
Here are some of the most important properties of inverse matrices:
1. The inverse of a matrix is unique.
If A is a matrix, then there can be only one possible inverse A−1 for A. To
understand why, suppose that A−1 is an inverse for A, and suppose that B is also
an inverse for A. Then
You should be able to understand each
step of this long equation.
B BI BAA−1 IA−1 A−1
so B must be the same matrix as A−1 .
2. The inverse of the inverse is the original matrix.
The relationship between A and A−1 is symmetric, so the inverse of A−1 is the
original matrix A.
A−1
Non-square matrices whose product is
the identity are sometimes called
one-sided inverses.
−1
A
3. Only square matrices can have inverses.
Though it is possible for the product of two non-square matrices to be the identity
matrix
"
#  −3 5 
"
#

1 −1 −1 
1 0
−3 3  2 −1 −3 
0 1
 −1 2 
if we reverse the order of multiplication we no longer get the identity
For square matrices, if the product AB is
the identity matrix, then BA is always the
identity as well, and so A and B are
inverses of one another.
 −3 5  "
 7 −2 −12 
#

 1 −1 −1
 3

−3
3
0 −6 

 2 −1 −3


 −1 2 
 3 −1 −5 
Only for square matrices can both products yields the identity matrix.
INVERSE MATRICES
2
4. Not every square matrix is invertible.
Some square matrices do not have an inverse. For example, the zero matrix
"
0
0
0
0
#
certainly has no inverse, since the product of the zero matrix with any other
matrix is always the zero matrix. But there are also less obvious examples, such
as the matrix
"
#
3 2
6 4
Sometimes invertible matrices are
referred to as non-singular.
which has no inverse. In general, a square matrix is called invertible if it has an
inverse, and singular if it does not.
5. The product of invertible matrices is invertible.
If A and B are invertible matrices, then the product AB is invertible as well, with
(AB ) −1 B −1 A−1
Note that the order of the two matrices is reversed when we take the inverse. This
order reversal is necessary to make the inverses cancel with AB when multiplied.
In particular,
ABB −1 A−1 AIA−1 AA−1 I
and
B −1 A−1 AB B −1 IB B −1 B I
Finding the Inverse
There is a simple formula for the inverse of a 2 × 2 matrix:
"
a
c
b
d
# −1
1
ad − bc
"
d
−c
#
−b
a
It is easy to check that this formula works. In particular,
1
ad − bc
"
d
−c
−b
a
#"
a
c
b
d
#
1
ad − bc
"
ad − bc
0
0
ad − bc
#
"
1
0
0
1
#
EXAMPLE 1
"
3
Find the inverse of the matrix
2
SOLUTION
#
8
.
6
According to the formula above, the inverse is
1
(3)(6) − (2)(8)
"
6
−2
−8
3
#
"
1
6
2 −2
−8
3
#
"
3
−1
−4
3/2
#
Incidentally, the denominator ad − bc in this formula is called the determinant of
the 2 × 2 matrix. A 2 × 2 matrix whose determinant is zero is not invertible.
INVERSE MATRICES
3
Finding the inverse of a 3 × 3 or larger matrix is much more difficult. For example,
the formula for the inverse of a 3 × 3 matrix is
 a
 d

g
b
e
h
c
f
i
 −1
1
 
ae i − a f h + b f g − bdi + cdh − ce g

 e i − f h

 f g − di

 dh − e g
ch − bi
b f − ce 
ai − c g
cd − a f 
b g − ah


ae − bd 
This formula is so complicated that it is rarely worthwhile to compute the inverse of a
3 × 3 matrix by hand, let alone larger sizes such as 4 × 4 or 5 × 5. Fortunately, most
modern calculators and computer algebra systems can compute the inverse of a matrix,
and several websites offer tools that let you perform matrix computations online.
Algebra of Inverse Matrices
Inverse matrices let you “divide” both sides of an equation by a matrix. For example, if
we have a matrix equation of the form
AX B
where X is an unknown matrix, we can solve for X by mutliplying both sides of the
equation by A−1 :
A−1 AX A−1 B
Since A−1 A I, the left side simplifies to just X, so
X A−1 B
EXAMPLE 2
Find a 2 × 2 matrix X so that
"
4
1
#
6
X 2
"
2
4
4
5
"
SOLUTION
Here
Multiplying through by the inverse of
"
4
1
6
2
−1 4
1
6
2
is the identity matrix, so
4
1
6
2
−1 4
1
4
1
6
2
# −1 "
#
4
1
6
X 2
"
#
#
4
1
4
1
6
gives
2
# −1 "
6
2
2
4
4
5
#
which simplifies to
"
6
X
2
is the same as IX, which is just X.
X But
"
4
1
6
2
# −1
so
6
2
1
(4)(2) − (6)(1)
"
X 4
1
1
−1/2
−3
2
#"
2
4
# −1 "
"
2
4
2
−1
4
5
#
4
5
#
−6
4
#
"
"
−10
7
1
−1/2
−11
8
−3
2
#
#
By the way, since the order of matrix multiplication matters, one must be careful to
multiply by A−1 in the same way on both sides of an equation. Given an equation
X Y
INVERSE MATRICES
4
we can either left-multiply by A−1 on both sides
A−1 X A−1 Y
or we can right-multiply by A−1 on both sides
XA−1 YA−1
However, we cannot left-multiply by A−1 on one side of the equation and right-multiply
by A−1 on the other side.
EXAMPLE 3
Make sure to understand the difference
between this example and the previous
one.
Find a 2 × 2 matrix X so that
"
X
4
1
6
2
#
"
2
4
4
5
#
"
This time we must right-multiply both sides by the inverse of
SOLUTION
"
X
4
1
6
2
#"
4
1
6
2
# −1
"
2
4
4
5
#"
4
1
6
2
4
1
#
6
. This gives
2
# −1
which simplifies to
"
X "
But
4
1
6
2
# −1
"
1
−1/2
2
4
4
5
#"
4
1
6
2
# −1
#
−3
, so
2
"
X 2
4
4
5
#"
1
−1/2
−3
2
#
"
0
3/2
2
−2
#
Solving Linear Systems with Inverses
Recall that any n × n linear system can be written in the form
Ax b
where A is the (square) matrix of coefficients, and b is the vector of constant terms. If
we know the inverse of A, we can solve this linear system by left-multiplying both sides
of the equation by A−1 . This gives
x A−1 b
Thus you can solve a linear system by finding the inverse of the matrix of coefficients.
This can be quite useful if you have several linear systems with the same coefficients
but different constant terms.
EXAMPLE 4
Solve each of the following linear systems:
3x + 2y 4
3x + 2y 1
3x + 2y 2
5x + 3y 7
5x + 3y 3
5x + 3y 5
INVERSE MATRICES
Each of these systems is really a vector equation:
SOLUTION
"
5
3
5
2
3
#"
x
y
#
" #
"
4
7
3
5
2
3
#"
#
x
y
" #
"
1
3
3
5
2
3
#"
x
y
#
" #
2
5
Thus the solutions are respectively
"
x
y
#
"
3
5
2
3
# −1 " #
"
3
5
2
3
# −1
#
"
2
−1
"
4
7
But
x
y
#
"
3
5
2
3
# −1 " #
"
3
−5
1
(3)(3) − (2)(5)
"
1
3
#
−2
3
x
y
#
"
"
−3
5
2
−3
"
x
y
#
"
3
5
2
3
# −1 " #
2
5
#
so the solutions are
"
x
y
#
"
x
y
#
"
3
−4
#
4
−5
#
EXERCISES
Find the inverse of the given matrix, or state that the matrix is not invertible.
1–4
"
1.
"
3.
#
5
9
2
3
3
4
−2
−3
"
2.
6
3
8
4
#
 3 0 0 


2 0


 0 0 5 
#
4.  0
Find scalars a, b, and c so that the given pair of matrices are inverses.
5–6
 1
2
3   −6
1
a 

 

5.  −2 −3 −3  ,  b −2 −3 

 

5
7   −1
c
1 
 3
 1
1
1 


6.  −3 −2 −6  ,


 −3 −1 −8 
 a
7

 −6 −5
 c −2
b 

3 

1 
7. Suppose that A, B, and C are invertible n × n matrices.
(a) Is the product ABC invertible? If so, what is its inverse?
(b) Is AB −1 C invertible? If so, what is its inverse?
Find a 2 × 2 matrix X that satisfies the given equation.
8–11
"
4
8. X
5
"
10.
8
5
7
9
#
#
"
"
6
1
X
4
0
1
2
2
1
3
1
#
#
"
9.
"
2
2
6
0
#
"
11.
#
1
2
0
X+
−1
3
4
5
X 7
#
"
"
3
2
2
2
1
4
#
"
5
6
2
5
#
"
2
5
1
4
#
#
3
X+
4
12. Use inverse matrices to find the solution to the each of the following linear systems:
4x + 5y 2
4x + 5y 8
4x + 5y 5
2x + 3y 6
2x + 3y 0
2x + 3y 1