The document discusses the inverse of matrices. It defines an inverse matrix A-1 as a matrix such that AA-1=I=A-1A, where I is the identity matrix. It provides methods for determining if a 2x2 matrix is invertible based on its determinant, and for finding the inverse of a matrix using elementary row operations. It also presents the Invertible Matrix Theorem, which provides equivalent conditions for a matrix to be invertible.

1. 2
2.2
© 2012 Pearson Education, Inc.
Math 337-102
Lecture 5
THE INVERSE OF A MATRIX

2. Slide 2.2- 2© 2012 Pearson Education, Inc.
Inverse Matrix - Definition
 An matrix A is said to be invertible if there is
an matrix C such that
and
where , the identity matrix.
 In this case, C is an inverse of A.
 This unique inverse is denoted by A-1
 A-1
A = I and AA-1
= I
 Invertible = nonsingular
 Not invertible = singular
n n×
n n×
CA I= AC I=
n
I I= n n×

3. Slide 2.2- 3© 2012 Pearson Education, Inc.
Inverse of 2x2 Matrix
 Theorem 4: Let . If , then
A is invertible and
If , then A is not invertible.
 The quantity is called the determinant of A,
and we write
 This theorem says that a matrix A is invertible
iff detA ≠ 0.
a b
A
c d
 
=  
 
0ad bc− ≠
1 1 d b
A
c aad bc
−
− 
=  −−  
0ad bc− =
ad bc−
det A ad bc= −
2 2×

4. Inverse of 2x2 Matrix - Example
 Find the inverse of A=
Slide 2.2- 4© 2012 Pearson Education, Inc.

5. Slide 2.2- 5© 2012 Pearson Education, Inc.
Solving Equations with Inverse Matrices
 Theorem 5: If A is an invertible matrix, then for
each b in Rn
, the equation Ax = b has the unique
solution x = A-1
b.
 Proof:
n n×

6. Solving Equations with Inverse Matrices - Example
 Use inverse matrix to solve:
3x1 + 4x2 = 3
5x1 + 6x2= 7
Slide 2.2- 6© 2012 Pearson Education, Inc.

7. Slide 2.2- 7© 2012 Pearson Education, Inc.
Theorem 2-6
a) If A is an invertible matrix, then A-1
is invertible and
(A-1
)-1
= A
b) If A and B are nxn invertible matrices, then so is
AB, and the inverse of AB is the product of the
inverses of A and B in the reverse order. That is,
(AB)-1
= B-1
A-1
a) If A is an invertible matrix, then so is AT
, and the
inverse of AT
is the transpose of A-1
. That is,
(AT
)-1
= (A-1
)T

8. Slide 2.2- 8© 2012 Pearson Education, Inc.
ELEMENTARY MATRICES
 An elementary matrix is one that is obtained by
performing a single elementary row operation on an
identity matrix.

9. Slide 2.2- 9© 2012 Pearson Education, Inc.
Theorem 2-7 – Method for Finding A-1
 An nxn matrix A is invertible iff A is row equivalent
to In. The row operations that reduces A to In also
transforms In into A-1
.

10. Slide 2.2- 10© 2012 Pearson Education, Inc.
ALGORITHM FOR FINDING
 Example 2: Find the inverse of the matrix
, if it exists.
 Solution:
A =
0 1 2
1 0 3
4 −3 8










1
A−

11. Slide 2.2- 11© 2012 Pearson Education, Inc.
ALGORITHM FOR FINDING
.
 Now, check the final answer.
1
9 / 2 7 3/ 2
2 4 1
3/ 2 2 1/ 2
A−
− − 
 = − −
 
−  
1
0 1 2 9 / 2 7 3/ 2 1 0 0
1 0 3 2 4 1 0 1 0
4 3 8 3/ 2 2 1/ 2 0 0 1
AA−
− −     
     = − − =
     
− −          
1
A−

12. Slide 2.3- 12© 2012 Pearson Education, Inc.
THE INVERTIBLE MATRIX THEOREM
Theorem 8: Let A be a square nxn matrix. Then the
following statements are equivalent. That is, for a given
A, the statements are either all true or all false.
a. A is an invertible matrix.
b. A is row equivalent to the nxn identity matrix.
c. A has n pivot positions.
d.The equation Ax = 0 has only the trivial solution.
e.The columns of A form a linearly independent set.
f.The linear transformation x⟼Ax is one-to-one.
g. Ax=b has at least one solution for each b in Rn
.

13. Slide 2.3- 13© 2012 Pearson Education, Inc.
THE INVERTIBLE MATRIX THEOREM (contd)
h. The columns of A span Rn
.
i. The linear transformation x⟼Ax maps Rn
onto Rn
.
j. There is an nxn matrix C such that CA=I.
k. There is an nxn matrix D such that AD=I.
l. AT
is an invertible matrix.

14. Slide 2.3- 14© 2012 Pearson Education, Inc.
THE INVERTIBLE MATRIX THEOREM
 The Invertible Matrix Theorem divides the set of all
nxn matrices into two disjoint classes:
 the invertible (nonsingular) matrices
 noninvertible (singular) matrices.
 Each statement in the theorem describes a property of
every nxn invertible matrix.
 The negation of a statement in the theorem describes
a property of every nxn singular matrix.
 For instance, an nxn singular matrix is not row
equivalent to In, does not have n pivot position, and
has linearly dependent columns.

15. Slide 2.3- 15© 2012 Pearson Education, Inc.
THE INVERTIBLE MATRIX THEOREM
 Example 1: Use the Invertible Matrix Theorem to
decide if A is invertible:
 Solution:
1 0 2
3 1 2
5 1 9
A
− 
 = −
 
− −  

16. Slide 2.3- 16© 2012 Pearson Education, Inc.
THE INVERTIBLE MATRIX THEOREM
 The Invertible Matrix Theorem applies only to square
matrices.
 For example, if the columns of a 4x3 matrix are
linearly independent, we cannot use the Invertible
Matrix Theorem to conclude anything about the
existence or nonexistence of solutions of equation of
the form Ax=b.

17. Slide 2.3- 17© 2012 Pearson Education, Inc.
INVERTIBLE LINEAR TRANSFORMATIONS
 Matrix multiplication corresponds to composition of
linear transformations.
 When a matrix A is invertible, the equation
can be viewed as a statement about linear
transformations. See the following figure.
1
x xA A−
=

18. Slide 2.3- 18© 2012 Pearson Education, Inc.
INVERTIBLE LINEAR TRANSFORMATIONS
 A linear transformation T:Rn
Rn
is invertible if there
exists a function S:Rn
Rn
such that
S(T(x)) = x for all x in Rn
T(S(x)) = x for all x in Rn
 Theorem 9: Let T:Rn
Rn
be a linear transformation
with standard matrix A. Then T is invertible iff A is an
invertible matrix.
T-1
:x⟼A-1
x