Matrix Inversion

Transpose Matris A m×n matris

A transpose matris AT, n×m matris

.

21

22221

11211

mnmm

n

n

aaa

aaaaaa

A

,

21

22212

12111

mnnn

m

m

T

aaa

aaaaaa

A

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

MATRİSLER

ÖDEV

5 * 1 = 5

13636

49

94

144

41

14

414

1001

3182

????

AB = BA = In.

1 AB

Properties of Matrix Inverse (A-1)-1 = A. (Ak)-1= (A-1)k (A-k ) (cA)-1 = (1/c)A-1, c ≠ 0. ( AT)-1 = (A-1)T. Ax = b isex = A-1b.

2-2 matrisin tersi

dcba

A

acbd

bcadA 11

31

82

determinant 6 – ( - 8) = 14

21

833182

acbd

bcadA 11

2183

141

1001

140014

141

6822242486

141

3182

2183

141

0))12(12(3264

3264

Tersi nedir?

3648

C

1001

3275

5273

:eg

1001

3275

5273

:eg

Tersi nedir?

acbd

bcadA 11

AA-1 = I

1131

21

21

23

21

3012

310

61

21

1001

)3(1)1(2)2(1)1(2)3(1)1(3)2(1)1(3

3211

1213

dcba

BveA5321

1001

5321

dcba

1001

535322

dbcadbca

1235153

02053

12

dandbcandadb

dbca

ca

4321

A

21

23

121324

211A

2)2(3)4(14321

10

22C

5162

A

21205

B

1 2 43 0 5

2 61 3

4 32 1

Minor ve kofactor minor Mij aij cofactor Cij = (-1)i+jMij.

Determinant

7687

G

14849)8(6)7(77687

Minors

835753542

835753542

835753542

11

5 7-61

3 -8M

12

-3 7-11

5 -8M

13

-3 5-34

5 3M

Minors

-61 -11 -3417 -41 26-53 29 -2

M

23

2 -426

5 3M

The resulting matrix of minors is:

835753542

Minor

-1 4A = ,

2 321 21

22 22

M = Minor of a = 4,M = Minor of a = -1

11 11

12 12

M = Minor of a = 3,M = Minor of a = 2

4 7 8A = -9 0 0 ,

2 3 4

M11 = Minor a11 =0 0

= =03 4

M23 = Minor a234 7

= =12-14=-22 3

M32 = Minor a324 8

= =0+72=72-9 0

Cofactors

Cofactors are the signed minors.The cofactor of element aij of matrix [A] is:

Therefore

The resulting matrix of cofactors is:

1 313 13

-1C M

-1 i jij ijC M

1 212 12

-1C M

1 111 11

-1C M

-61 11 -34-17 -41 -26-53 -29 -2

C

Kofactor

C11 = Cofactor a11 = (–1)1 + 1 M11 = (–1)1 +1 0 0=0

3 4

C23 = Cofactor a23 = (–1)2 + 3 M23 = 4 7

22 3

C32 = Cofactor a32 = (–1)3 + 2M32 =4 8

- =-72-9 0

4 7 8A = -9 0 0

2 3 4

Determinant 2-2 Matris

dcba

A bcadAA )det(

11 12

21 22

a aA =

a a

|A| = = a11a22 – a21a12

a a

a a

11 12

21 22

4 -3determinant :

2 5 4 -3= 4×5-2× -3 =20+6 =26

2 5

Determinant 3-3 Matris

ihgfedcba

A

)()(

)det(

idbhfageccdhbfgaeicegbfgcdhbdiafhaei

hged

cigfd

bihfe

aAA

2 3 - 5

1 - 2 7 - 2 7 17 1 - 2 = 2 - 3 + -5

4 1 -3 1 -3 4-3 4 1

hged

cigfd

bihfe

aihgfedcba

23028

)6(5)12(0)14(2)1(0)2(35)5(0)4(30)5(2)4(12

2013

54053

04251

2420513

502

420513

502

214321112

1391014

)9(1)10(1)7(2)2(4)1(11)3(4)2(11)3(1)2(22

1421

12431

)1(2132

2214321112

Matris Tersi

adjoint matrix?

)(11 AadjA

A

1011

1011

tersi

1001

1011

1011

1011

1011

Matrix Inversion1 2 3

1 2 3

1 2 3

2 4 5 36-3 5 7 75 3 8 -31

x x xx x x

x x x

2 -4 5-3 5 75 3 -8

A

Matrix Inversion

1

2

3

2 -4 5 36-3 5 7 75 3 -8 -31

xxx

A x B

The set of equations in matrix form is:

1x A B

Adjoint matrixadjoint matrix [A],Adj[A]transpose cofactor matrix [A].

-61 11 -34-17 -41 -26-53 -29 -2

C

determinant

-61 -17 -5311 -41 -29-34 -26 -2

adj A

2 -4 5

-3 5 7 -336

5 3 -8

A

Tadj A C

Matris Tersi 1 1A adj A

A

1

-61 -17 -53-336 -336 -33611 -41 -29

-336 -336 -336-34 -26 -2

-336 -336 -336

A

1-61 -17 -53

1 11 -41 -29-336

-34 -26 -2A

1

61 17 53336 336 336-11 41 29336 336 33617 13 1168 168 168

A

SARRUS

4.2 3 86 7 14 5 9

4.2 3 86 7 14 5 9

2 36 74 5

-224+10

+162

= -52

-126 +12+240

126 - (-52)126 + 52= 178

=126

5.5 1 22 3 53 2 3

-18 +50

+6 = 38

45 - 15 + 8 = 38

38 - 38= 0

5 12 33 2

0

24031

11

2

1

xx

Sonuçmatrisi

Sabitler

03240

yxyx

BilinmeyenlerX ~ x1Y ~ x2

47

1. x & yax + by = mcx + dy = n

2. 2x2 matris tersi

3. x,y cözümü

nm

yx

dcba

acbd

bcad1

dcba 1

nm

acbd

bcad1

yx

Cramer's

2 -3D = = 2+9 =11 0

3 1 1

7 -3D = =7+15=22

5 1

2

2 7D = =10-21=-11

3 5

1 2

D 0D D22 -11By Cramer's Rule x= = =2 and y= = =-1D 11 D 11

2x-3y=7,3x+y=5

49

3x +4y = 55x = 7-6y

x+7y = 1.243y -x = 0.76

8x = 3y -1x+y =-7

71

yx

1138

76.0

24.1yx

3171

75

yx

6543

21

42

21

21252830

21

75

3546

21

yx

x = -1 y = 2

52

5522

111

561211

111

71

8131

111

yx

2.016.0

26.1

101

76.024.132.572.3

101

76.024.1

1173

101

yx

x = -0.16 y = 0.2

x = -2 y = -5

Linear Denklem Sistemleri (Cramer’s)

2 2 2 2a x+b y+c z = d ... ii

1 1 1 1a x+b y+c z = d ... i

3 3 3 3a x+b y+c z = d ... iii

31 2 DD Dx = , y = z= D 0,

D D D,

1 1 1 1 1 1 1 1 1

2 2 2 1 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3

a b c d b c a d cD= a b c , D = d b c , D = a d c

a b c d b c a d c

1 1 1

3 2 2 2

3 3 3

a b dD = a b d

a b d

5x - y+ 4z = 52x + 3y+ 5z = 25x - 2y + 6z = -1

5 -1 4D= 2 3 5

5 -2 6

1

5 -1 4D = 2 3 5

-1 -2 6

= 5(18+10)+1(12+5)+4(-4 +3)= 140 +17 –4= 153

= 5(18+10) + 1(12-25)+4(-4 -15)= 140 –13 –76 =140 - 89= 51 0

3

5 -1 5D = 2 3 2

5 -2 -1

= 5(-3 +4)+1(-2 - 10)+5(-4-15)= 5 – 12 – 95 = 5 - 107= - 102

1 2

3

D 0D D153 102By Cramer's Rule x= = =3, y= = =2D 51 D 51

D -102and z= = =-2D 51

2

5 5 4D = 2 2 5

5 -1 6

= 5(12 +5)+5(12 - 25)+ 4(-2 - 10)= 85 + 65 – 48 = 150 - 48= 102