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© 2012 Pearson Education, Inc.

Math 337-102Lecture 5

THE INVERSE OF A MATRIX

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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-1A = I and AA-1 = I Invertible = nonsingular Not invertible = singular

n n×n n×

CA I= AC I=nI I= n n×

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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 bA

c d

=

0ad bc− ≠

1 1 d bA

c aad bc− −

= −− 0ad bc− =ad bc−det A ad bc= −

2 2×

Inverse of 2x2 Matrix - Example

Find the inverse of A=

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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-1b.

Proof:

n n×

Solving Equations with Inverse Matrices - Example

Use inverse matrix to solve:

3x1 + 4x2 = 3

5x1 + 6x2 = 7

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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-1A-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

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ELEMENTARY MATRICES An elementary matrix is one that is obtained by

performing a single elementary row operation on an identity matrix.

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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.

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ALGORITHM FOR FINDING Example 2: Find the inverse of the matrix

, if it exists.

Solution:

A =0 1 21 0 34 −3 8

1A−

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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−

− − = − − =

− −

1A−

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THE INVERTIBLE MATRIX THEOREMTheorem 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.

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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.

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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.

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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

− = − − −

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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.

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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− =

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INVERTIBLE LINEAR TRANSFORMATIONS

A linear transformation T:RnRn is invertible if there exists a function S:RnRn such that

S(T(x)) = x for all x in Rn

T(S(x)) = x for all x in Rn

Theorem 9: Let T:RnRn be a linear transformation

with standard matrix A. Then T is invertible iff A is an invertible matrix.

T-1:x⟼A-1x