lesson 1 linear algebra

8/3/2019 Lesson 1 Linear Algebra

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LESSON 1

LINEAR ALGEBRA



3

Matrix Algebra

What is matrix?



In this lecture, we will lear n about the tools for

solving linear systems of equations

There are several methods we have lear ned beforesuch as solving by graphical method,by using

substitution and elimination method

and etc.



Content

Matrix

Determinant

± Diagonal Method ± Cofactors Method

Elementary Row Operations

In

verse Matrix



¹¹¹

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º

¸

©©©

©©

ª

¨

!v

mnm2m1

2n2221

1n1211

nm

aaa

aaa

aaa

A

.

/.//

-

.

What is Matrix A rectangular array of real (or complex)

numbers arranged in m rows and n columns

Dimension

of the matrix

A

Column

Row



Some Special MatricesSquare matrix

¹¹

º

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ª

¨ !

1.70.4

2.61.2D

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2105.0

413.4

15.10

E

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¨!

100

010

001

I3

Identity matrix, nI

¹¹ º

¸©©ª

¨!

10

01I2



8

Basic oper ations of matrix

(+, -, x, T)



9

Matrix Addition

A is a m x n matrix and B is a m x n matrix

C = A + B

where for all i=1,2,«,m ; j=1,2,«,nijijij bac !

¹¹¹

º

¸

©©©

ª

¨

!

333231

232221

131211

aaa

aaa

aaa

A

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¸

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!

333231

232221

131211

bbb

bbb

bbb

B

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¸

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!¹¹¹

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¸

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!

333332323131

232322222121

131312121111

333231

232221

131211

bababa

bababa

bababa

ccc

ccc

ccc

C



10

Example

Notice that matrices with different dimension cannot

be added together:

No valid result for A + B

¹¹ º

¸©©ª

¨

!

420

113 A

¹¹ º

¸©©ª

¨

!

420

113 A ¹¹

º

¸©©ª

¨

!

6310

162 B

¹¹ º ¸©©

ª¨

!¹¹

º ¸©©

ª¨

!

10510

075

6432100

116123 B A

¹¹ º

¸©©ª

¨

!

22

13 B



11

Matrix Subtr action

A is a m x n matrix and B is a m x n matrix

C = A - B

where for all i=1,2,«,m ; j=1,2,«,nijijij bac !

¹¹¹

º

¸

©©©

ª

¨

!

333231

232221

131211

aaa

aaa

aaa

A

¹¹¹

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¸

©©©

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!

333231

232221

131211

bbb

bbb

bbb

B

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¨

!¹¹¹

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¸

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!

333332323131

232322222121

131312121111

333231

232221

131211

bababa

bababa

bababa

ccc

ccc

ccc

C



12

Example

Notice that matrices with different dimension cannot

be subtracted together:

No valid result for A - B

¹¹ º

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¨

!

420

113 A

¹¹ º

¸©©ª

¨

!

420

113 A ¹¹

º

¸©©ª

¨

!

6310

162 B

¹¹ º ¸©©

ª¨

!¹¹

º ¸©©

ª¨

!

2110

251

6432100

116123

)( B A

¹¹ º

¸©©ª

¨

!

22

13 B



13

Scalar Multiplication

A is a m x n matrix and E is a scalar (real number)

C = E A

where for all i=1,2,«,m ; j=1,2,«,n

e.g. C = A

ijij ac vE!

¹¹¹

º

¸

©©©

ª

¨

vEvEvE

vEvEvE

vEvEvE

!¹¹¹

º

¸

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!

333231

232221

131211

333231

232221

131211

aaa

aaa

aaa

ccc

ccc

ccc

C

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¨

!¹¹¹

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vvv

vvv

vvv

!

987

654

321

901080107010

601050104010

301020101010

...

...

...

C ¹¹¹

º

¸

©©©

ª

¨

!

908070

605040

302010

AE !



14

Matrix Multiplication

A is a m x n matrix and B is a n x p matrix

C = A X B ,where C is a m x p matrix

Where

for all i=1,2,«,m ; j=1,2,«,p

To find the element cij (i-th row and j-th column of C = AB),we have to multiply each element in the i-th row of A by thecorresponding element in the j-th column of B and addthem together.

§!

!!n

k

k jik n jin ji jiij babababac1

2211 .



15

Matrix Multiplication

C11 from 1st row of A and 1st column of B

[a11 a12 a13«a1m] 1st row of A

[b11 b21 b31«bm1] 1st column of B

C11= a11 b11 + a12 b21 + a13 b31 +«+ a1m bm1



16

Example

A is a 2 x 3 matrix and B is a 3 x 3 matrix

C = A X B, expect C is a 2 x 3 matrix

Find determine C, elements by elements

C11= a11 b11 + a12 b21 + a13 b31

= 1x1 + 2x0 + 3x(-1)

= -2

¹¹ º

¸©©ª

¨!

123

321 A

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¸

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¨

!

231

120

201

B ¹¹ º

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¨!!

232221

131211

ccc

ccc ABC



17

Example

C21= a21 b11 + a22 b21 + a23 b31 = 3x1 + 2x0 + 1x(-1)

= 2

C12= a11 b12 + a12 b22 + a13 b32 = 1x0 + 2x2 + 3x3

= 13

¹¹ º

¸©©ª

¨!

123

321 A

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¸

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ª

¨

!

231

120

201

B ¹¹ º

¸©©ª

¨!!

232221

131211

ccc

ccc ABC

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¨

! 123

321 A

¹¹

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º

¸

©©

©

ª

¨

!231120

201

B ¹¹ º

¸

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!! 232221

131211

ccc

ccc

ABC



18

Example

C22= a21 b12 + a22 b22 + a23 b32 = 3x0 + 2x2 + 1x3

= 7

C13= a11 b13 + a12 b23 + a13 b33 = 1x(-2) + 2x1 + 3x2

= 6

¹¹ º

¸©©ª

¨!

123

321 A

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¸

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ª

¨

!

231

120

201

B ¹¹ º

¸©©ª

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232221

131211

ccc

ccc ABC

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! 123

321 A

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!231120

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B ¹¹ º

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!! 232221

131211

ccc

ccc

ABC



19

Example

C23= a21 b13 + a22 b23 + a23 b33 = 3x(-2) + 2x1 + 1x2

= -2

Plug in the computed elements back to C, we have

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123

321 A

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231

120

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B ¹¹ º

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ccc

ccc ABC

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272

6132 ABC



20

Example

What is AxB?

What is BxA?

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vvvv

vvvv!

01

00

01001100

00001000 AB

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00 A ¹¹

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01

00 B

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vvvv!

00

00

10010001

10000000 BA

A B B A v{vMultiplicatio

norder is importa

nt!



21

Tr anspose

The transpose of a m x n matrix A, denoted

by = B, is the n x m matrix where

E.g.

jiij ab !

TA

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¨! 654

321A

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!63

52

41T

A

321!B

¹

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!

3

2

1T

B



DeterminantsIf A is square matrix then the determinant

function is denoted by det and det(A) is defined

to be the sum of all the signed elementary

matrices of A.

nnn2n1

2n2221

1n1211

aaa

aaa

aaa

.

/.//

-

.

The result of a

determinant isa singlenumber.

!! AAdet



Finding Determinants

³D

ia

gona

l Meth

od´Determinant function for a 2×2 matrix.

2221

1211

aaaa!! AAdet

2211aa!

1221aa



Example 1

Compute the det(A) for the following matrix.

¹¹ º

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! 24

53

A

24

53Adet ! 26! 206 !




³D

ia

gona

l Meth

od´Determinant function for a 3×3 matrix.

!B

332211aaa!

132231aaa

333231

232221

131211

aaa

aaa

aaa

3231

2221

1211

aa

aa

aa

312312aaa

322113aaa

112332aaa

122133aaa



Example 2


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111

420

022

A



Example 2«solution

!B

4! 0

1-11

4-2-0

02-2

11

2-0

2-2

8 0 8- 0!B 20



Minor & Cof actor If A is a square matrix then the minor of ,

denoted by , is the determinant of the

submatrix that results from removing the i th row

and j th column of A.

jia

ji

ji

ji M1C

!

jiM

If A is a square matrix then the cof actor of ,

denoted by , is the number

jia

jiC



Example 3

For the following matrix compute the minor, M23

and the cofactor, C23

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!

740

259

613

A



Example 3«solution

!740259

613

M 32

! 40

13

!32C

12

!

321

32

M1 12

12!

Minor

Cofactor




³Meth

od of Cof a

ctors´

nnn2n1

2n2221

1n1211

aaa

aaa

aaa

.

/.//

-

.

!A

If A is an n×n matrix.

Choose any row, then:

nini2i2i1i1i Ca.....CaCa !A

Choose any column, then:

jn jn j2 j2 j1 j1 Ca.....CaCa !A

cof actor



Example 4


¹¹¹

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!

111

420

022

A



Example 4«solution

2!1142

04262 !

Choose first row, then:

312111 C0C2C2 !A 312111 M0M-2M2 !

20A !

0

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¸

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¨

!

111

420

022

A

2

11

40



Elementary Row Oper ations

Elementary row operations are manipulations

that can be performed on the row of a matrix.

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43

21

Three basic elementary operations.

1. Interchange any two rows.1. Interchange any two rows.

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21

4321 R R !



Elementary Row Oper ations

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4321

2. Multiply any row by a non2. Multiply any row by a non--zerozero

scalar.scalar.

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434211

R 2R !

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¨

43

21

3. Add k times of any row to another 3. Add k times of any row to another

row.row.

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¨

43

85211 R R 2R !



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º

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¨ 31

Example 5«solution

¹¹ º

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¨ 42

31

122 2R R R! ¹¹ º

¸

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¨ 31

100

10

10062:2R

42:R

1

2

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10R R 22

z!10

10

10

10

0:

10

R 2

211 3R R R !

01

30:3

31:

2

1

R

R

01

10



Inverse Matrix

If A is square matrix and we can find

another matrix of the same size B, such

that

Then we call A invertible and we say that

B is an inverse of the matrix A

nIABBA !!



Remarks on Inverse Matrix

Inverse matrix is only for square matrix

The inverse for a matrix is unique

The inverse of a matrix A is denoted as A-1

If A is invertible, then A is a non-singular

matrix

If A is not invertible, then A is a singular matrix



¹¹ º

¸

©©ª

¨

Finding Inverse Matrix

³Using Row Oper a

tion´Inverse for a 2×2 matrix.

operation

r owelementary

Find row operations that will convert the

first 2 columns into I2.

1

A

2I

The third and fourth columns should then

contain A-1.

¹¹ º

¸

©©ª

¨22A v 2I



Example 6«solution

Form a new matrix

¹

¹

º

¸

©

©

ª

¨

10

01

42

31Matrix

A

122 2R R R! ¹¹

º

¸©©

ª

¨ 0131

12100

12-100

0262:2R

1042:R

1

2

10

R

22R

! ¹¹ º

¸©©ª

¨ 0131

10

1

5

110 101

51

101

102

1010

10-0

10

R

10

:2



Example 6«solution

¹¹ º

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¨

10

1

5

110

¹¹ º

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¨

!

101

51

103

52

1A

211 3R R R !01

10

3

5

2

103

52

103

53

2

1

01

30:3R

0131:R

The inverse of matrix A is



¹¹ º

¸

©©ª

¨

Finding Inverse Matrix³Using Row Oper ation´

Inverse for a 3×3 matrix.

operation

r owelementary

Find row operations that will convert the

first 3 columns into I3.

1

A

3I

The last three columns should then contain

A-1.

¹¹ º

¸

©©ª

¨33A v 3I



Finding Inverse Matrix³Using Row Oper ation´

Please refer the attachment



Finding Inverse Matrix³Using Adjoint Cof actor´


If matrix

is invertible, its inverse will be

¹¹ º ¸©©

ª¨!

dc baA

¹¹ º

¸©©ª

¨

!

ac

bd

A

1A 1



Example 7

Determine the inverse for the following

matrix.

¹¹ º

¸

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¨

! 24

53

A

¹¹ º ¸©©

ª¨v!1

A

45

26 32

A

1¹¹ º ¸©©

ª¨

!26

3

13

2

26

5

13

1



Finding Inverse Matrix³Using Adjoint Cof actor´


If matrix

is in

vertible, its inverse will be

¹¹¹

º

¸

©©©

ª

¨

!

333231

232221

131211

A

aaa

aaa

aaa

Aad jointA

1A

1 v!



For Example please refer the attachment

lesson 1 linear algebra

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