Linear Equation and Matrices

8/12/2019 Linear Equation and Matrices

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1

CHAPTER 1

Systems of Linear Equations and

Matrices


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prepared by Razana Alwee

2


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(values s1,s2,..snare substitute for x1,x2,.xn)


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Solution


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Example

Determine whether either of the points (1,5)

and (0,2) is a solution to the given system of

equations.

y= 3x2

y=x6


8

To check the given possible solutions, just plug the x-and y-coordinates into the equations, and check to

see if they work.

How?


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Solution

checking (1,5):

(5) = 3(1)2

5 =325 =5 (solution checks)

(5)=(1)6

5 = 16

5 =5 (solution checks)

Since the given point works in each equation, it is a

solution to the system.


9


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(unique or infinite solutions)


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Unique Solution

A system,

The point where the lines cross {x=3.6, y=0.4} is thesolution that satisfies both equations simultaneously.

The solution is unique and consistent.

13


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Consistent


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Infinite Many Solutions

A systems,

This system is redundant because the second equation

is equivalent to the first one.

They 'cross' at an infinite number of points, so there are

an infinite number of solutions

Consistent


15


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(Consistent)


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No Solutions

A system

consists of two parallel lines that never cross. Thus

there is no solution.

Inconsistent 17

(1)

(2)


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(Inconsistent)


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Solving Linear System

Matrices are helpful in rewriting a linear system in a very

simple form.

Matrix form, AX= B


19

mnmm

n

n

aaa

aaa

aaa

A

:

:::::

:

21

22221

11211

nx

x

x

X:

2

1

nb

b

b

B:

2

1

,


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A matrix is said to be row-echelon form if the following

conditions are satisfied:

Every row with all 0 entries is below every row with

nonzero entriesEach leading entry of a row is in a column to the right of the

leading entry of the row above it(leading entry-any nonzero

value)

(The second row starts with more zero than the first(left to

right)All entries in a column below a leading entry are zeros.

Row-echelon Matrices


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24

0 0 0 0 0 0

0 0 0 0 0 #

0 0 # * * *

# * * * * *#-leading entry(left most

non zero entry)*-any number/values including 0

Row-echelon Matrices


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26

(0s below and

above each

leading 1) 0 0 0 0 0 00 0 0 0 0 10 0 1 * * 0# * 0 * * 0

Reduced Row-echelon Matrices


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or row-echelon form


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Elementary Row Operation

Elementary row operations can be used to transform thematrix into its row-echelon form. If we denote row 1 byR1, row 2 by R2, etc., and ais a scalar, the threeelementary row operations are as follows:

1) Swap two rows (equations), denoted R1R2 (this wouldswap rows 1 and 2)

2) Multiply a row by a nonzero scalar, denoted aR1 (thiswould multiply row 1 by k)

3) Add a multiple of a row to another row, denoted R1 +aR2 (this would add ktime row 2 to row 1, and replacethe previous row 1)


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Example

Given is a linear equation system,

Augmented form

124

423

72

221

321

321

xxx

xxx

xxx

1

47

241

123112


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Example (cont.)

Swap two rows

14

7

241123

112R

1 R

3

74

1

112123

241


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Multiply row 1 with a =3 = 3R1and add the multiple row

to another row, R2

122 3RRR

7)1(34

7)2(31

14)4(32

0)1(33

7

71

112

7140241

3 122 RRR

7

4

1

112

123

241

The notation means to take row 2 and subtract 3 times

row 1 from it to produce the new augmented matrix.

Example (cont.)


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Bi Bi

Bi Bi

Bi +Bi Bj


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Solution


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Solution (cont.)


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Solution (cont.)


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Solution (cont.)


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Solution (cont.)


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Solution (cont.)


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X=(3,2,1)T

Solution (cont.)


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Solution


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Solution (cont.)


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Solution (cont.)


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(solve leading variables using back-substitutions)

(assign nonleading variables as parameter)

Solution (cont.)


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50


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Introduction to Matrix

A rectangular arrangement of number is called a matrix

Matrices are usually denoted using upper case letter A,

B, C, K, P, etc

example

A =

2 0 -4

3 1 7 B =

5 2 8

3 4 3

3 9 6


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The size of matrix is depend on the

number of rows and columns

A matrix with mrows and ncolumns is

called an m-by-nmatrix (m x nmatrix).

Example:

A =

2 0 -4

3 1 7

rows

columns

2

3

2 x 3 matrix


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A matrix of size 1 x nis called a row

matrix:

A matrix of size mx 1 is called a

column matrix:

C = -1 5 2

B =1

4


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54

A matrix with equal numbers of rows and

columns is called a square matrix:

E =7 3

8 -2C =

4 12 6

7 0 4

11 3 -5



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Example

C =

2 0

3 1B =

5 2 8

7 4 33 9 6 -4 9

D =

6

2-4

E =4 -5

8 1F = 6 2 9


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The numbers in the matrix are called its entries

The entry in the ith rowand jth column of

matrixXis referred as the (i,j)-entry of thematrixXand denoted by a lower case letter

with two subscripts indices,xi,j

If Ais an m x nmatrix. The (i,j)-entry of Ais

denoted by ai,j

A =a1,1 a1,2 a1,3 a1,n..a2,1 a2,2 a2,3 a2,n..

: : : :am,1 am,2am,3 am,n..


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Example: Ais an 2 x 3 matrix

A =

2 0 -4

3 1 7

The (1,2) entry of A is 0; a1,2= 0The (2,3) entry of A is 7;a2,3= 7

B =

5 2 8

7 4 3

3 9 6

The (3,1) entry of B is ?;

b1,3= ?


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Exercise

What is the (3,4)-entry?

9006489

5030323

344569

15350

X=


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Matrices

The diagonal entries in an mxnmatrix A=[aij] are a11,a22 ,a33and they form the main diagonalof A.

mnmmm

n

n

ij

aaaa

aaaa

aaaa

aA

..

:::::

..

..

321

2232221

1131211


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Matrices

A diagonal matrix is a square matrix whose

nondiagonal entries are zero.

Example: nxnidentity matrixIn

1000

0100

0010

0001

nI


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Matrices

The mxnmatrix whose entries are all zero is called the

zero matrixand will be denoted by 0.

0000

0000

0000

0000


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Matrices

Two matrices A and B are equal (A=B)if they have the

same number of rows and columns and if corresponding

entries are equal.


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Example

Given

Discuss the possibility that A=B, A=C and B=C

713

402A

13

02

B

dc

baC


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Example

A=B(impossible because A and B are different

size)

A=C(impossible because A and C are different

size)

B=C (possible provided that corresponding

entries are equal)

713402A

13

02B

dc

baC


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Matrix Addition

If A and B are 2 matrices of the same size,

A + B is defined to be the matrix of the same size formed

by adding corresponding entries.

If A=[aij] and B=[bij],[aij]+[bij]=[aij+bij]


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Matrix Addition

A =a1,1 a1,2 a1,3a2,1 a2,2 a2,3

B =b1,1 b1,2 b1,3b2,1 b2,2 b2,3

A+B=a1,1 a1,2 a1,3a2,1 a2,2 a2,3

b1,1+b2,1+

b1,2+

b2,2+

b1,3+b2,3+


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Matrix Subtraction

A-B is defined by subtracting corresponding entries.

If A=[aij] and B=[bij],

[aij]-[bij]=[aij- bij]


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Matrix Subtraction

A =a1,1 a1,2 a1,3a2,1 a2,2 a2,3

B =b1,1 b1,2 b1,3b2,1 b2,2 b2,3

A-B=a1,1 a1,2 a1,3a2,1 a2,2 a2,3

b1,1-b2,1-

b1,2-

b2,2-

b1,3-b2,3-


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Example

415

023A

357

241B

1412

262

34)5(175

2042)1(3BA

762

224

34)5(175

2042)1(3BA


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Theorem

If A, B and C denote arbitrary mxn

matrices, then,

A+B=B+A (commutative law)A+(B+C)=(A+B)+C (associative law)

0+A=A (0 is the mxn zero matrix)

A+(-A)= 0


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Scalar Multiplication

If Ais a matrix and cis a number, the scalar product

cAis the matrix formed from Aby multiplying each entry

of Aby the number c.

If A=[aij] ; cA=c[aij] a1,1 a1,2 a1,3A = a2,1 a2,2 a2,3

cA =ca1,1ca1,2ca1,3ca2,1ca2,2ca2,3

cA =a1,1 a1,2 a1,3a2,1 a2,2 a2,3

c


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Example

415

023A

357

241B

8210046

41502322A

171711

689

8210

046

91521

612323 AB

91521

6123

357

24133B


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Theorem

Let Aand Bdenote matrices and let c

and ddenote numbers,

c(A+B)=cA+cB(c+d)A=cA+dA

c(dA)=(cd)A


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Theorem

If cA=0, then either c=0 or

A=0

c(0)=0

0A=0


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Matrix Multiplication

If Ais an mxnmatrix and Bis an nxpmatrix

The number of columns of the left matrix (n)

must be same as the number of rows of the right

matrix (n)

Their matrix product (AB) is the mxpmatrix

whose (i,j) entry is the dot product of row iof A

and columnjof B.


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Matrix Multiplication

Matrix of size 2x3Matrix of size 3x2

zyx

yx

z

y

x

45

23

415

023

Matrix of size 2x3 Matrix of size 3x1 Matrix of size 2x1

cba

ba

zyx

yx

c

b

a

z

y

x

45

23

45

23

415

023

Matrix of size 2x2


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Transposition

Given an mxn matrix A, the transposeof

Ais the nxmmatrix, denoted by AT, whose

columns are formed from the

corresponding rows of A.


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Example

415

023A

40

12

53TA

zy

xwB

zx

ywBT

3

2

1

TC

321C

Row

Column


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Theorem

Let Aand Bdenote matrices whose

sizes are appropriate for the following

sums and products.(AT)T=A

(A+B) T=AT+BT

For any scalar r, (rA)T=rAT

(AB)T=BTAT


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Exercise

315

012A

8117

204B

31

40

12

C

(a) A + B (b) 3A-B (c) AC (d) A-2B

(e) AT+C (f) CB (g) AT+BT (h) CT-B

(i) AAT (j) ATA (k) ATB (l) CT-A


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Determinants

The determinant of a 2x2 matrix

For 1x1 matrix A=[a], |A|=a

bcaddcbaAA

det

dc

baA


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Example

Compute the determinant of

45

12A

3585*14*245

12detdet

A


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Submatrix

For any square matrix A, let Aijdenote the submatrix

formed by deleting the i-th row and j-th column of A.

Example:

987654

321

A

97

64

12A

C f


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Cofactors

Given A=[aij]

The(i,j)-cofactor of Ais Ci,jgiven by

ij

ji

ij AC det)1(

The + or sign in the

(i,j)-cofactor dependson the position of aijinthe matrix.

(-1)1+1


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Determinants of n 2

nn CaCaCaA 1112121111 ...det

D i f 2


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Determinants of n 2

For n2, the determinant of an nn

matrixA=[aij] is the sum of n terms of the

form a1jdet A1j, with plus and minussigns alternating, where the entries a11,

a12, ,a1nare from the first row of A.

AaAaAaA nn det)1(..detdet 1

112121111

D t i t


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Determinants

The determinant of an nn matrix A can

be computed by a cofactor expansion

across any row or down any column.

D t i t


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Determinants

The expansion across the i-th row using the cofactor

The cofactor expansion down the j-th column is

ininiiii CaCaCaA ...det 2211

njnjjjjj CaCaCaA ...det 2211

D t i t


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Determinants

The determinant of a 3x3 matrix using first row

expansion:

333231

232221

131211

aaa

aaa

aaa

A

3231

2221

13

3331

2321

12

3332

2322

11 detdetdetdet

aa

aaa

aa

aaa

aa

aaaA

E l


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Example

Use a cofactor expansion across the 3rd row to compute

det A.

l


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example

E i


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Exercise

M t i I


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Matrix Inverses

If Ais a square matrix, a matrix Cis

called an inverse of Aif,

AC= I and CA=IC=A-1

M t i I


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Matrix Inverses

E l


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Example

Matri In erses (3 3)


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Matrix Inverses (3 x 3)

Using cofactor matrix to compute the inverse:

TAA

AA

A )trixcofactorma(||

1)adjoint(||

11

E l


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Example

Det (A) =|A| =

A=Find the inverse of the matrix

Solution:

1. Find the determinant of A (in this case, using Row 1)

Example(cont )


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Example(cont.)

A12

A13=

A11= (-1+2) =1

A21=

A22=

A23=

A31=

A32=

A33=

2. Find the determinant of

submatrix (minor of entry ai j):

Example(cont )


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Example(cont.)

3. Find the adjoint of matrix A:

TAA )trixcofactorma(Adjoint

3. Find the cofactors of matrix A:

141

183

121

cofactorsofmatrix

141

183

121

111

482

131

Example(cont )


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100

A-1 =

4. Calculate the inverse:

Example(cont.)

)adjoint(||

11A

A

A

Matrix Inverses using Linear System


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Matrix Inverses using Linear System

Suppose Ais a square matrix and there exists a

sequence of elementary row operations that

carry AI. Then Ais invertible and this same

sequence carries IA-1.

[ A I] [ I A-1]

where the row operations on A and Iare carried out

simultaneously.

Example


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Example

Solution:

Solution (cont )


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Solution (cont.)

Solution (cont )


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Solution (cont.)

Solution (cont )


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Solution (cont.)

Solution (cont )


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Solution (cont.)

Solution (cont )


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Solution (cont.)

Exercise


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Exercise

Find the inverse of the given matrices.

042

361

123

A

i) Using the Adjoint

i) Using the linear systems.

Matrix Factorization


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The solution of a system of linear equations AX = B

can be computed much more quickly if the matrix A

can be factored in the form

A = LU

where

L is a lower triangular matrix

Uis an upper triangular matrix



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a a a

a a a

a a a

x

x

x

b

b

b

11 12 13

21 22 23

31 32 33

1

2

3

1

2

3

a a a

a a a

a a a

l

l l

l l l

u u u

u u

u

11 12 13

21 22 23

31 32 33

11

21 22

31 32 33

11 12 13

22 23

33

0 0

0 0

0 0

(Ax=B)

(A=LU)



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l u l u l u

l u l u l u l u l u

l u l u l u l u l u l u

a a a

a a a

a a a

11 11 11 12 11 13

21 11 21 12 22 22 21 13 22 23

31 11 31 12 32 22 31 13 32 23 33 33

11 12 13

21 22 23

31 32 33

l

l l

l l l

u u u

u u

u

a a a

a a a

a a a

11

21 22

31 32 33

11 12 13

22 23

33

11 12 13

21 22 23

31 32 33

0 0

0 0

0 0



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AX=Bcan be solved in 2 stages:

First solve LY=Bfor Yby forward substitution.

Then solve UX=YforXby back substitution.



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LY=B

Forward substitution

33

2321313

3

22

1212

2

11

1

1

,

,

l

ylylby

l

ylby

l

by

l

l l

l l l

y

y

y

b

b

b

11

21 22

31 32 33

1

2

3

1

2

3

0 0

0



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UX=Y

Back substitution

u u u

u u

u

x

x

x

y

y

y

11 12 13

22 23

33

1

2

3

1

2

3

0

0 0

11

31321211

22

32322

33

33

,

,

u

xuxuyx

u

xuyx

u

yx

Doolittle Factorization


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The diagonal elements of matrix Lare 1s.

nn

n

n

nnnnnn

n

n

u

uuuuu

ll

l

aaa

aaaaaa

ULA

..00

:..::

..0

..

1..

:..::

0..10..01

..

:..::

..

..222

11211

21

21

21

22221

11211



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nnnn

n

n

nn

n

n

nn aaa

aaa

aaa

u

uu

uuu

ll

l

..

:..::

..

..

..00

:..::

..0

..

1..

:..::

0..1

0..01

21

22221

11211

222

11211

21

21

Example


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Example

Find the LUfactorization of the matrix

using Doolittle form and use it to solve the

linear system.

14

11

8

241

113

121

3

2

1

x

x

x

Solution (cont.)


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Solution (cont.)

241

113

121

00

0

1

01

001

33

2322

131211

3231

21

u

uu

uuu

ll

l

111u 212 u 113 u

Solution (cont.)


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Solution (cont.)

241

113

121

00

0

121

1

01

001

33

2322

3231

21

u

uu

ll

l

313

21 l 11131 l

Solution (cont.)


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( )

241

113

121

00

0

121

11

013

001

33

2322

32 u

uu

l

5)2)(3(122 u 2)1)(3(123 u

Solution (cont.)


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( )

241

113

121

00

250

121

11

013

001

3332 ul

5

2

5

)2)(1(432

l

Solution (cont.)


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( )

241

113

121

00

250

121

11

013

001

3352 u

51)2(

52)1)(1(233

u

Solution (cont.)


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( )

241113

121

00250

121

11013

001

51

52

Solution (cont.)


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( )

14

11

8

11

013

001

3

2

1

52 y

y

y81y

13)8)(3(112 y

54

52

3 )13)(()8)(1(14 y

TY 54

138

Solution (cont.)


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( )

54

3

2

1

51

13

8

00

250

121

x

x

x 451

54

3 x

15

)4)(2(132 x

21

)4)(1()1)(2(8

1

x TX 412

Exercise


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Find the LUfactorization of the matrixAusing Doolittle

form and use it to solve the linear systemAX=B.

(a) B=[7 2 10]T (b) B=[23 35 7]T

321

921

611

A

Crout Factorization


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The diagonal elements of matrix Uare 1s.

ll l

l l l

u uu

a a aa a a

a a a

11

21 22

31 32 33

12 13

23

11 12 13

21 22 23

31 32 33

0 00

10 1

0 0 1

Example


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Find the LUfactorization of the matrix using Crout form

and use it to solve the linear system.

26

20

12

611

352

327

3

2

1

x

x

x

Solution


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333231

2221

11

0

00

lll

ll

l

L

100

10

1

23

1312

u

uu

U

Solution (cont.)


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611

352327

100

101

000

23

1312

333231

2221

11

uuu

lll

lll

LU=A

Solution (cont.)


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711 l 221 l 131 l

611

352

327

100

10

1

0

00

23

1312

333231

2221

11

u

uu

lll

ll

l

Solution (cont.)


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7

212u 7

313

u

611

352

327

100

10

1

1

02

007

23

1312

3332

22 u

uu

ll

l

Solution (cont.)


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7

31

7

22522 l

31

15

27

33

7

31

23

23

u

u

611

352

327

100

10

7/37/21

1

02

007

23

3332

22 u

ll

l

Solution (cont.)


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7

9

7

2

132

l 31

192

31

)15(

7

)9(

7

3

633

l

611

352

327

100

31/1510

7/37/21

1

07/312

007

3332 ll

Solution (cont.)


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611

352

327

100311510

73721

31192791

07312

007

Solution (cont.)


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26

20

12

31192791

07312

007

3

2

1

y

y

y

192

902

,31

116

,7

12

3

2

1

y

y

y

Solution (cont.)


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192902

31116712

100

311510737/21

3

2

1

x

xx

192

138

192

902

7

3

192

282

7

2

7

12

192

282

192

902

31

15

31

116

192902

1

2

3

x

x

x

Exercise


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Find the LUfactorization of the matrixAusing Crout

form and use it to solve the linear systemAX=B.

(a) B=[-4 10 5]T (b) B=[20 49 32]T

231

351

642

A

Linear Equation and Matrices

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