Algebra w3b

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Algebra of Matrices

Real Numbers

(Solve for x)

mn Matrices

(Solve for X)

x + a = b

X + A = B

x + a + (-a) = b + (-a)

X + A + (-A) = B + (-A)

x + 0 = b – a

X + 0 = B - A

x = b – a

X = B - A

matrix
equation

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Algebra of Matrices

Real Numbers

(Solve for x)

mn Matrices

(Solve for X)

x

a = b

X A = B

x

a

a

-1

= b

a

-1

XAA

-1

= BA

-1

x = b

a

-1

X = X E = B A

-1

matrix
equation

/A

-1

R

A X = B

A

-1

AX = A

-1

B

X = E X= A

-1

B

/A

-1

L

A

B

X

!!!

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Definition 7.3:

Definition 7.3:

A square matrix A is called noninvertible (or
singular) when det(A)=0. W przeciwnym
wypadku, matrix A is called nonsingular
(invertible).

Definition 8.3:

Definition 8.3:

An quadratic matrix A (of order n) is nonsingular
(invertible) if there exist a quadratic matrix B
such that
AB = BA = E

n

where E

n

is the identity matrix of order n. The

matrix B is called the (multiplicative) inverse of
A.

!!!

• Nonsquare Matrix DO NOT HAVE inverses

A

nm

· B

mn

B

mn

·

A

nm

quadratic Matrix
of order n

quadratic matrix
of order m

• Not all square matrices possess inverses

V. Matrices – The Inverse of a Matrix

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Theorem 1.

Theorem 1.

5

5

:

:

If A is an invertible matrix, then its inverse is

unique. We denote the inverse of A by A

-1

.

Uniqueness of an Inverse Matrix

Proof:

A is invertible  it has at least one inverse

Suppose that B and C are inverses of A

AB = BA = EAC = CA = E

AB = E

C(AB) = CE

(CA)B = C

EB = C

B = C= A

-1

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How to Find an Inverse Matrix ?

by Gauss-Jordan

Elimination

(elementary row

operations are used)

by Its Adjoint

(determinants

are applied)

Finding the Inverse of Matrix

Example

:

Show that B is the inverse of A :

,

1

1

2

1

,

1

1

2

1

B

A

1

0

0

1

1

2

1

1

2

2

2

1

1

1

2

1

1

1

2

1

AB

1

0

0

1

1

2

1

1

2

2

2

1

1

1

2

1

1

1

2

1

BA

1

A

B

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!!!

Finding the Inverse of Matrix by Gauss-Jordan Elimination

E

A

1

A

E

Elementary row

operations

NOTE:

If A cannot be row reduced to E, then A is singular

Example

:

Find the inverse of the following matrix

3

2

6

1

0

1

0

1

1

A

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1

0

0

0

1

0

0

0

1

3

2

6

1

0

1

0

1

1

E

A

write the n 2n matrix that consists of given matrix A

on the left and the nn identity matrix E on the right to

obtain [A : E]. We call this process adjoining the
matrices A and E.

If possible, row reduce A to E using elementary row
operations on the entire matrix [A : E]. The result will be
the matrix [E : A

-1

]. If this is not possible, then A is not

invertible.

1

0

0

0

1

0

0

0

1

3

2

6

1

0

1

0

1

1

)

6

(

1

)

1

(

1

r

r

1

0

6

0

1

1

0

0

1

3

4

0

1

1

0

0

1

1

~

)

4

(

2

)

1

(

2

r

r

Algorithm (

Gauss-Jordan Elimination

)

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1

4

2

1

3

1

1

3

2

1

0

0

0

1

0

0

0

1

~

1

4

2

1

3

3

1

3

2

1

A

Check your work by multiplying to see A

-1

A = A A

-1

= E

1

0

0

0

1

0

0

0

1

1

4

2

1

3

3

1

3

2

3

2

6

1

0

1

0

1

1

1

4

2

0

1

1

0

1

0

1

0

0

1

1

0

1

0

1

~

)

1

(

1

4

2

0

1

1

0

1

0

1

0

0

1

1

0

1

0

1

~

)

1

(

3

)

1

(

3

r

r

1

0

0

0

1

0

0

0

1

3

2

6

1

0

1

0

1

1

1

4

2

1

3

3

1

3

2

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Example

:

Find the inverse of the following matrix

2

3

2

2

1

3

0

2

1

A

1

0

0

0

1

0

0

0

1

2

3

2

2

1

3

0

2

1

E

A

)

6

(

1

)

1

(

1

r

r

1

0

2

0

1

3

0

0

1

2

7

0

2

7

0

0

2

1

~

)

1

(

2 

r

1

1

1

0

1

3

0

0

1

0

0

0

2

7

0

0

2

1

~

A portion” of the
matrix has a row of
zeros

It is not posible to
rewrite the matrix
[A :
E] to [E : A

-1

]

A has no
inverse

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Finding the Inverse of Matrix by Its Adjoint

Definition 2.5:

Definition 2.5:

The

The

adjoint

adjoint

of matrix A is denoted

of matrix A is denoted

adj

adj

(A) and has

(A) and has

folowing form

folowing form

Where

Where

is a

is a

cofactor

cofactor

, and

, and

is a minor of the element .

is a minor of the element .

 

T

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

adj

nn

n

n

n

n

n

n

3

2

1

3

33

32

31

2

23

22

21

1

13

12

11

A

 

 

ij

j

i

ij

C

A

det

1

 

ij

A

det

ij

a

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Find the

adjoint of

,

2

0

1

1

2

0

2

3

1

A

Example

:

,

2

0

1

1

2

0

2

3

1

A

 

4

2

0

1

2

det

1

1

1

11





C

Continuing this process produces the following matrix of cofactors of A

2

1

7

3

0

6

2

1

4

2

0

3

1

1

0

2

1

1

2

2

3

0

1

3

1

2

1

2

1

2

0

2

3

0

1

2

0

2

1

1

0

2

0

1

2

2

3

2

1

0

1

7

6

4

)

(A

adj

Matrix of cofactors

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Theorem 2.5:

Theorem 2.5:

If A is an nn invertible matrix, then

)

(

)

det(

1

A

A

A

1

adj

Example

:

Find the inverse of the following matrix

2

3

2

2

1

3

0

2

1

A

Proof: without proof

2

3

2

2

1

3

0

2

1

det

)

det(A

0

12

6

8

2

matrix A is

singular !!!

 A

-1

does not exist

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3

2

6

1

0

1

0

1

1

A

Example

:

1

3

2

6

3

2

6

1

0

1

0

1

1

det

Find the inverse of matrix

A is invertible

3

2

6

1

0

1

0

1

1

A

2

3

2

1

0

)

1

(

1

1

11

C

T

adj

2

)

(A

background image

3

2

6

1

0

1

0

1

1

A

Example

:

1

3

2

6

3

2

6

1

0

1

0

1

1

det

Find the inverse of matrix

A is invertible

3

2

6

1

0

1

0

1

1

A

3

3

6

1

1

)

1

(

2

1

12

C

T

adj

2

)

(A

T

adj

3

2

)

(A

background image

3

2

6

1

0

1

0

1

1

A

Example

:

1

3

2

6

3

2

6

1

0

1

0

1

1

det

Find the inverse of matrix

A is invertible

3

2

6

1

0

1

0

1

1

A

2

2

6

0

1

)

1

(

3

1

13

C

T

adj

3

2

)

(A

T

adj

2

3

2

)

(A

background image

3

2

6

1

0

1

0

1

1

A

Example

:

1

3

2

6

3

2

6

1

0

1

0

1

1

det

Find the inverse of matrix

A is invertible

3

2

6

1

0

1

0

1

1

A

3

3

2

0

1

)

1

(

1

2

21

C

T

adj

2

3

2

)

(A

T

adj

 3

2

3

2

)

(A

background image

3

2

6

1

0

1

0

1

1

A

Example

:

1

3

2

6

3

2

6

1

0

1

0

1

1

det

Find the inverse of matrix

A is invertible

3

2

6

1

0

1

0

1

1

A

3

3

6

0

1

)

1

(

2

2

22

C

T

adj

 3

2

3

2

)

(A

T

adj

3

3

2

3

2

)

(A

background image

T

adj

3

3

2

3

2

)

(A

3

2

6

1

0

1

0

1

1

A

Example

:

1

3

2

6

3

2

6

1

0

1

0

1

1

det

Find the inverse of matrix

A is invertible

3

2

6

1

0

1

0

1

1

A

23

C

4

2

6

1

1

)

1

(

3

2

T

adj

4

3

3

2

3

2

)

(A

background image

T

adj

4

3

3

2

3

2

)

(A

3

2

6

1

0

1

0

1

1

A

Example

:

1

3

2

6

3

2

6

1

0

1

0

1

1

det

Find the inverse of matrix

A is invertible

3

2

6

1

0

1

0

1

1

A

31

C

1

1

0

0

1

)

1

(

1

3

T

adj

1

4

3

3

2

3

2

)

(A

background image

T

adj

1

4

3

3

2

3

2

)

(A

3

2

6

1

0

1

0

1

1

A

Example

:

1

3

2

6

3

2

6

1

0

1

0

1

1

det

Find the inverse of matrix

A is invertible

3

2

6

1

0

1

0

1

1

A

1

1

1

0

1

)

1

(

2

3

32

C

T

adj

1

1

4

3

3

2

3

2

)

(A

background image

T

adj

1

1

4

3

3

2

3

2

)

(A

3

2

6

1

0

1

0

1

1

A

Example

:

1

3

2

6

3

2

6

1

0

1

0

1

1

det

Find the inverse of matrix

A is invertible

3

2

6

1

0

1

0

1

1

A

1

0

1

1

1

)

1

(

3

3

33

C

T

adj

1

1

1

4

3

3

2

3

2

)

(A

1

4

2

1

3

3

1

3

2

1

4

2

1

3

3

1

3

2

1

1

1

A


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