The Inverse of a Matrix
S. F. Ellermeyer
June 12, 2009
These notes closely follow the presentation of the material given in David
C. Lay’s textbook Linear Algebra and its Applications (3rd edition). These
notes are intended primarily for in-class presentation and should not be regarded as a substitute for thoroughly reading the textbook itself and working
through the exercises therein.
The Inverse of a Square Matrix
Each n
n identity matrix
2
6
6
In = 6
4
3
0
0
1
0 7
7
.. . . .. 7
. . 5
.
0 0
1
1
0
..
.
plays a role in matrix algebra similar to the role played by the number 1 in
the regular algebra of numbers. In particular, if C is any m n matrix, then
CIn = C, and if D is any n m matrix, then In D = D.
In the regular algebra of numbers, every real number a 6= 0 has a
unique multiplicative inverse. This means that there is a unique real number,
a 1 such that aa 1 = a 1 a = 1. For example the multiplicative inverse of 5
is 1=5 (which we also denote by 5 1 ) because 5 5 1 = 5 1 5 = 1.
We will ask this same type of questions for square matrices: Given an
n n matrix, A, can we …nd an n n matrix B such that AB = BA = In ?
We begin by giving some de…nitions that apply to matrices that are
not necessarily square.
1
De…nition 1 If A is an m n matrix and C is a n
CA = In , then C is said to be a left inverse of A.
m matrix such that
De…nition 2 If A is an m n matrix and D is a n
AD = Im , then D is said to be a right inverse of A.
m matrix such that
It was proved in homework problems 23–25 in Section 2.1 that if a
matrix A has both a left and a right inverse, then A must be a square matrix
and the left and right inverses of A must be equal to each other. In other
words: If A has size m n, C and D both have size n m, CA = In , and
AD = Im , then m = n and C = D.
Considering what has been said in the above paragraph, it only makes
sense to talk about a matrix having (or not having) both a left and a right
inverse if the matrix is square. However, it is possible for a non–square matrix
to have a left inverse but no right inverse (or vice-versa).
2
De…nition 3 A matrix that has both a left and a right inverse is said to be
an invertible matrix.
Example 4 The matrix
1 2
3 4
A=
is invertible because the matrix
B=
2
3
2
is both a left and a right inverse of A.
3
1
1
2
Theorem 5 If a matrix, A, is invertible, then A has a unique left inverse
and a unique right inverse and these left and right inverses are equal to each
other. (We call this unique matrix the inverse of A and denote it by A 1 .)
Proof. Suppose that B is a left inverse of A. Since A is invertible, we
know that A also has a right inverse. However, we also know that every
right inverse of A must be equal to B. In other words, B can be the only
right inverse of A. But this means that every left inverse of A must equal B.
Thus B is the only left inverse of A. We have proved that A has a unique
left inverse. By similar reasoning, we can prove that A has a unique right
inverse. It is also clear (from the reasoning in homework problems 23-25 of
Section 2.1) that these left and right inverses must be equal to each other.
Example 6 Fort the matrix
A=
1 2
3 4
,
the matrix
B=
2
3
2
1
1
2
is the only left inverse of A, and B is also the only right inverse of A. The
matrix B is called the inverse of the matrix A and we can write A 1 = B.
4
Theorem 7
1. If A is an invertible matrix, then A
matrix, and
1
A 1
= A:
1
is also an invertible
2. If A and B are invertible matrices of the same size, then AB is also
an invertible matrix and
(AB)
1
= B 1A 1.
3. If A is an invertible matrix, then AT is an invertible matrix and
AT
1
5
= A
1 T
.
Elementary Matrices
An elementary matrix is a matrix that can be obtained from an identity
matrix by performing a single elementary row operation.
Example 8 The matrices E1 ; E2 ; and E3 shown below are all
matrices.
2
3
2
3
2
1 0 0
1 0 0
1
2
4
5
4
5
4
E1 = 0 0 1 , E2 = 0 3 0 , E3 = 0 1
0 1 0
0 0 1
0 0
elementary
3
0
0 5.
1
Since every elementary row operation is reversible, all elementary
matrices are invertible and their inverses are obtained by performing the
reverse elementary row operation on the identity matrix.
6
Example 9 To obtain the elementary
2
1
4
E1 = 0
0
matrix
3
0 0
0 1 5,
1 0
we interchange rows 1 and 2 of the identity matrix. Thus, to obtain E1 1 , we
interchange rows 1 and 2 of the identity matrix. Therefore,
2
3
1 0 0
E1 1 = 4 0 0 1 5 .
0 1 0
To obtain the elementary matrix
2
3
1 0 0
E2 = 4 0 3 0 5 ,
0 0 1
we scale row 2 of the identity matrix by a factor of 3. Thus, to obtain E2 1 ,
we scale row 2 of the identity matrix by a factor of 1=3. Therefore,
2
3
1 0 0
E2 1 = 4 0 13 0 5 .
0 0 1
To obtain the elementary matrix
2
1
4
E3 = 0
0
3
2 0
1 0 5,
0 1
we replace row 1 of the identity matrix by (row 1 + (-2 times row 2)). Thus,
to obtain E3 1 , we replace row 1 of the identity matrix by (row 1 + (2 times
row 2)).. Therefore,
2
3
1 2 0
E3 1 = 4 0 1 0 5 .
0 0 1
7
Lemma 10 Suppose that B is a matrix obtained by performing a single elementary row operation on the matrix A. Also, suppose that E is the elementary matrix obtained by performing this same elementary row operation on
I. Then B = EA.
Example 11 The matrix
2
1
4
B = 11
1
3
4
2
3
6
1 5
2
is obtained by replacing row 2 of the matrix
2
3
1
3 6
3 5
A=4 9 0
1
2
2
with (row 2 plus (2 times row 3)).
The elementary matrix
2
3
1 0 0
E=4 0 1 2 5
0 0 1
is obtained by replacing row 2 of the identity matrix with (row 2 plus (2 times
row 3)).
Observe that B = EA.
8
Theorem 12 An n n matrix, A, is invertible if and only if A~In . In this
case, any sequence of elementary row operations that transforms A into In
also transforms In into A 1 .
9
An Algorithm for Finding A
1
To …nd the inverse of an invertible n
1. Form the matrix
n matrix A:
A In
2. Perform elementary row operations on A In until A has been
transformed into In . The result will be In A 1 .
Example 13 Use the algorithm described above to …nd the inverse of the
matrix
3
2
1
1
1 0
6 0
1
0 1 7
7.
A=6
4 1
2
4 0 5
1
0
0 0
10
The Inverse of a 2
2 Matrix
Theorem 14 If
A=
a b
c d
then A is invertible if and only if ad
A
1
=
bc 6= 0. If A is invertible, then
1
ad
,
d
c
bc
b
a
Example 15 Let
A=
Then ad
bc = (1) (4)
A
1
(2) (3) =
=
1
2
4
3
1 2
3 4
.
2 6= 0 and
2
1
11
=
2
3
2
1
1
2
.