Download The Inverse of a Square Matrix

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
In 
1
0

0
0
1

0
   
0
0

1
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 CI n  C, and if D is any
n  m matrix, then I n D  D.
In the regular algebra of numbers, every real number a  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 find an n  n matrix B such that AB  BA  I n ?
We begin by giving some definitions that apply to matrices that are not necessarily
square.
Definition If A is an m  n matrix and C is a n  m matrix such that CA  I n , then C is
said to be a left inverse of A.
Definition If A is an m  n matrix and D is a n  m matrix such that AD  I m , then D is
said to be a right inverse of A.
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  I n , and AD  I m , 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).
1
Definition A matrix that has both a left and a right inverse is said to be an invertible
matrix.
Example The matrix
1 2
A
3 4
is invertible because the matrix
B
2
1
3
2
 12
is both a left and a right inverse of A.
2
Theorem 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 Fort the matrix
A
1 2
3 4
,
the matrix
B
2
1
3
2
 12
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.
3
Theorem
1.
If A is an invertible matrix, then A 1 is also an invertible matrix, and
1
A 1   A.
2.
If A and B are invertible matrices of the same size, then AB is also an
invertible matrix and
AB 1  B 1 A 1 .
3.
If A is an invertible matrix, then A T is an invertible matrix and
T
1
A T   A 1  .
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Elementary Matrices
An elementary matrix is a matrix that can be obtained from an identity matrix by
performing a single elementary row operation.
Example The matrices E 1 , E 2 , and E 3 shown below are all elementary matrices.
1 0 0
E1 
0 0 1
0 1 0
1 2 0
1 0 0
, E2 
0 3 0
0 0 1
, E3 
0 1 0
.
0 0 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.
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Example To obtain the elementary matrix
1 0 0
E1 
0 0 1
,
0 1 0
we interchange rows 1 and 2 of the identity matrix. Thus, to obtain E 1
1 , we
interchange rows 1 and 2 of the identity matrix. Therefore,
1 0 0
E 1
1 
0 0 1
.
0 1 0
To obtain the elementary matrix
1 0 0
E2 
0 3 0
,
0 0 1
we scale row 2 of the identity matrix by a factor of 3. Thus, to obtain E 1
2 , we scale
row 2 of the identity matrix by a factor of 1/3. Therefore,
1 0 0
E 1
2 
0
1
3
0
.
0 0 1
To obtain the elementary matrix
1 2 0
E3 
0 1 0
,
0 0 1
we replace row 1 of the identity matrix by (row 1  (-2 times row 2)). Thus, to
obtain E 1
3 , we replace row 1 of the identity matrix by (row 1  (2 times row 2))..
Therefore,
1 2 0
E 1
3

0 1 0
.
0 0 1
6
Lemma 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 The matrix
1 3 6
B
11 4 1
1 2 2
is obtained by replacing row 2 of the matrix
1 3 6
A
9 0
3
1 2 2
with (row 2 plus (2 times row 3)).
The elementary matrix
1 0 0
E
0 1 2
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.
7
Theorem An n  n matrix, A, is invertible if and only if A~I n . In this case, any sequence
of elementary row operations that transforms A into I n also transforms I n into A 1 .
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An Algorithm for Finding A 1
To find the inverse of an invertible n  n matrix A:
1.
Form the matrix
2.
Perform elementary row operations on
A In
transformed into I n . The result will be
A In
I n A 1
until A has been
.
Example Use the algorithm described above to find the inverse of the matrix
A
1
1
1 0
0
1
0 1
1 2 4 0
1
0
.
0 0
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The Inverse of a 2  2 Matrix
Theorem If
A
a b
,
c d
then A is invertible if and only if ad  bc  0. If A is invertible, then
A 1 
d b
1
ad  bc
c a
Example Let
A
1 2
3 4
.
Then ad  bc  14  23  2  0 and
A 1  1
2
4 2
3 1

2
1
3
2
 12
.
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