Konsep Matriks

102
Konsep Matriks

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MATRIKS. Konsep Matriks. MATRIX. Concept of Matrix. Macam-macam Matriks. Kompetensi Dasar : Mendeskripsikan macam-macam matriks Indikator : Matriks ditentukan unsur dan notasinya Matriks dibedakan menurut jenis dan relasinya. Kinds of Matrix. Basic Competences : - PowerPoint PPT Presentation

Transcript of Konsep Matriks

Konsep Matriks

Concept of Matrix

Hal.: 3 Matriks Adaptif

Macam-macam Matriks

Kompetensi Dasar :Mendeskripsikan macam-macam matriks

Indikator :1. Matriks ditentukan unsur dan notasinya2. Matriks dibedakan menurut jenis dan

relasinya

Hal.: 4 Matriks Adaptif

Kinds of Matrix

Basic Competences :Describing the kinds of matrix

Indicators :1. Matrix is determined by its elements and

notations2. Matriks matrix is distinguished by its kinds

and relations

Hal.: 5 Matriks Adaptif

Pengertian Matriks Matriks adalah susunan bilangan-bilangan yang terdiri

atas baris-baris dan kolom-kolom.

a11 a12…….a1j ……a1n

a21 a22 ……a2j…….a2n : : : :ai1 ai2 ……aij…….. ain

: : : :am1 am2……amj……. amn

A = baris

kolom

Notasi: Matriks: A = [aij]Elemen: (A)ij = aij

Ordo A: m x n

Masing-masing bilangan dalam matriks disebut entri atau elemen. Ordo (ukuran) matriks adalah jumlah baris kali jumlah kolom.

Macam – macam Matriks

Hal.: 6 Matriks Adaptif

Definition of Matrix Matrix is the arrangement of numbers

which consists of rows and columns.

a11 a12…….a1j ……a1n

a21 a22 ……a2j…….a2n : : : :ai1 ai2 ……aij…….. ain

: : : :am1 am2……amj……. amn

A = rows

column

Notation: Matrix: A = [aij]Element: (A)ij = aij

Order A: m x n

Each of the numbers in matrix is called as entry or element. Order (size) of matrix is the value of the row number multiplied by the number of column.

Kinds of Matrix

Hal.: 7 Matriks Adaptif

Macam-macam Matriks

Matriks baris adalah matriks yang hanya terdiri dari satu baris.

21A 2 5

31xB

41xC

1 -8 25

-2 0 14 8

1. Matriks Baris

Hal.: 8 Matriks Adaptif

Kinds of Matrix

Row matrix is a matrix which consists of one row.

21A 2 5

31xB

41xC

1 -8 25

-2 0 14 8

1. Row matrix

Hal.: 9 Matriks Adaptif

Macam-macam Matriks

12P

2. Matriks Kolom

Matriks Kolom adalah matriks yang hanya terdiri dari satu kolom

2

-7

13Q9

2

1

Hal.: 10 Matriks Adaptif

Kinds of Matrix

12P

2. Column matrix

Column matrix is a matrix which consists of one column.

2

-7

13Q9

2

1

Hal.: 11 Matriks Adaptif

1 2 42 2 23 3 3

3. Matriks PersegiMatriks persegi (bujur sangkar) adalah matriks yang jumlah baris dan jumlah kolom sama.

Trace(A) = 1 + 2 + 3

Trace dari matriks adalah jumlahan elemen-elemen diagonal utama

diagonal utama

Macam – macam Matriks

Hal.: 12 Matriks Adaptif

1 2 42 2 23 3 3

3. Square matrixSquare matrix is a matrix which has the same numbers of rows and columns.

Trace(A) = 1 + 2 + 3

Trace from matrix is the total numbers from the main diagonal elements.

Main diagonal

Kinds of Matrix

Hal.: 13 Matriks Adaptif

4. Matriks Nol

Matriks nol adalah matriks yang semua elemennya nol

0 0 0 0 00 0

1 00 1

1 0 00 1 00 0 1

1 0 0 00 1 0 00 0 1 0 0 0 0 1

I2I3 I4

Matriks identitas adalah matriks persegi yang elemen diagonal utamanya 1 dan elemen lainnya 0

Macam- macam Matriks

Hal.: 14 Matriks Adaptif

4. Zero matrix

zero matrix is a matrix which all of its elements are zero.

0 0 0 0 00 0

1 00 1

1 0 00 1 00 0 1

1 0 0 00 1 0 00 0 1 0 0 0 0 1

I2I3 I4

Matrix identity is a square matrix which its main diagonal element is 1 and the other element is 0.

Kinds of Matrix

Hal.: 15 Matriks Adaptif

5. Matriks ortogonalMatriks A orthogonal jika dan hanya jika AT = A –1

0 -1

1 0A =

0 1

-1 0AT=

B = ½√2 -½√2

½√2 ½√2 BT= ½√2 ½√2

-½√2 ½√2

Jika A adalah matriks orthogonal, maka (A-1)T = (AT)-1

= A-1

= B-1

(A-1)T = (AT)-1 A-1 AT

Macam-macam Matriks

Hal.: 16 Matriks Adaptif

5. Orthogonal MatrixMatrix A is orthogonal if and only if AT = A –1

0 -1

1 0A =

0 1

-1 0AT=

B = ½√2 -½√2

½√2 ½√2 BT= ½√2 ½√2

-½√2 ½√2

If A is orthogonal matrix, so (A-1)T = (AT)-1

= A-1

= B-1

(A-1)T = (AT)-1 A-1 AT

Kinds of Matrix

Hal.: 17 Matriks Adaptif

Macam – macam Matriks

Definisi:Transpose matriks A adalah matriks AT, kolom-kolomnya adalah baris-baris dari A, baris-barisnya adalah kolom-kolom dari A.

4 2 6 7

5 3 -9 7A = AT = A’ =

4 5

2 3

6 -9

7 7

Jika A adalah matriks m x n, maka matriks transpose AT berukuran ………..

[AT]ij = [A]ji

n x m

Hal.: 18 Matriks Adaptif

Kinds of Matrix

Definisi:Transpose matrix A is matrix AT, its columns are rows of A, its rows is columns of A.

4 2 6 7

5 3 -9 7A = AT = A’ =

4 5

2 3

6 -9

7 7

if A is matrix m x n, so matrix transpose AT should be ………..

[AT]ij = [A]ji

n x m

Hal.: 19 Matriks Adaptif

Kesamaan dua matriks Dua matriks sama jika ukuran sama dan setiap entri yang bersesuaian sama.

1 2 4

2 1 3A =

1 2 4

2 1 3B =

1 2 2

2 1 3C =

2 1 2

2 1 3D =

1 2 4

2 2 2E =

x 2 4

2 2 2F =

2 2 2

4 5 6

9 0 7

G = H =? ? ?

? ? ?

? ? ?

A = B

C ≠ D

E = F jika x = 1

G = H

2 2 2

4 5 6

9 0 7

Macam – macam Matriks

Hal.: 20 Matriks Adaptif

Similarity of two matrixes Two matrix are similar if its size is similar and each symmetrical entry is similar

1 2 4

2 1 3A =

1 2 4

2 1 3B =

1 2 2

2 1 3C =

2 1 2

2 1 3D =

1 2 4

2 2 2E =

x 2 4

2 2 2F =

2 2 2

4 5 6

9 0 7

G = H =? ? ?

? ? ?

? ? ?

A = B

C ≠ D

E = F jika x = 1

G = H

2 2 2

4 5 6

9 0 7

Kind of Matrix

Hal.: 21 Matriks Adaptif

Matriks Simetri

Matriks A disebut simetris jika dan hanya jika A = AT

4 2

2 3A =

4 2

2 3A’ = A simetri

1 2 3 42 5 7 0

3 7 8 2 4 0 2 9

A = = AT

Macam-macam Matriks

Hal.: 22 Matriks Adaptif

Symmetrical matrix

Matrix A is called symmetric if and only if A = AT

4 2

2 3A =

4 2

2 3A’ = A symmetric

1 2 3 42 5 7 0

3 7 8 2 4 0 2 9

A = = AT

Kinds of Matrix

Hal.: 23 Matriks Adaptif

Sifat-sifat transpose matriks

A AT (AT)T

(AT )T = A1. Transpose dari A transpose adalah A:

4 2 6 7

5 3 -9 7

4 5

2 3

6 -9

7 7

4 5

2 3

6 -9

7 7

= A

Contoh:

Macam-macam Matriks

Hal.: 24 Matriks Adaptif

properties of transpose matrix

A AT (AT)T

(AT )T = A1. Transpose of A transpose is A:

4 2 6 7

5 3 -9 7

4 5

2 3

6 -9

7 7

4 5

2 3

6 -9

7 7

= A

Example:

Kinds of Matrix

Hal.: 25 Matriks Adaptif

Macam-macam Matriks

2. (A+B)T = AT + BT

A+B

(A+B)T

T

BT

B

T

A

T

AT

=

=

+

+

Hal.: 26 Matriks Adaptif

Kinds of Matrix

2. (A+B)T = AT + BT

A+B

(A+B)T

T

BT

B

T

A

T

AT

=

=

+

+

Hal.: 27 Matriks Adaptif

Macam-macam Matriks

3. (kA)T = k(A) T untuk skalar k

kA

(kA)T = k(A)T

A

T T

k

Hal.: 28 Matriks Adaptif

Kinds of Matrix

3. (kA)T = k(A) T for scalar k

kA

(kA)T = k(A)T

A

T T

k

Hal.: 29 Matriks Adaptif

Macam-macam Matriks

4. (AB)T = BT AT

(AB)T

AB

T T

AB

T

=

AB = BTAT

Hal.: 30 Matriks Adaptif

Kinds of Matrix

4. (AB)T = BT AT

(AB)T

AB

T T

AB

T

=

AB = BTAT

Hal.: 31 Matriks Adaptif

Macam-macam Matriks

Isilah titik-titik di bawah ini1. A simetri maka A + AT= ……..2. ((AT)T)T = …….3. (ABC)T = …….4. ((k+a)A)T = ….....5. (A + B + C)T = ……….

Kunci:1. 2A 2. AT

3. CTBTAT 4. (k+a)AT 5. AT + BT + CT

Soal :

Hal.: 32 Matriks Adaptif

Kind of Matrix

Fill in the blanks bellow1. A symmetric then A + AT= ……..2. ((AT)T)T = …….3. (ABC)T = …….4. ((k+a)A)T = ….....5. (A + B + C)T = ……….

Answer keys:1. 2A 2. AT

3. CTBTAT 4. (k+a)AT 5. AT + BT + CT

Quiz :

Hal.: 33 Matriks Adaptif

OPERASI MATRIKS

Kompetesi DasarMenyelesaikan Operasi Matriks

Indikator1. Dua matriks atau lebih ditentukan hasil

penjumlahan atau pengurangannya2. Dua matriks atau lebih ditentukan hasil kalinya

Hal.: 34 Matriks Adaptif

OPERATION OF MATRIX

Basic competenceFinishing operation matrix

Indicator1. Two or more matrixes is defined by the result of

their addition or subtraction 2. Two or more matrixes is defined by the result of

their multiplication

Hal.: 35 Matriks Adaptif

Penjumlahan dan pengurangan dua matriksContoh :

10 22

1 -1A = 2 6

7 5B =

10+2 22+6

1+7 -1+5A + B =

12 28

8 4=

8 16

-6 -6= A - B = 10-2 22-6

1-7 -1-5

OPERASI MATRIKS

Hal.: 36 Matriks Adaptif

Addition and subtraction of two matixesExample:

10 22

1 -1A = 2 6

7 5B =

10+2 22+6

1+7 -1+5A + B =

12 28

8 4=

8 16

-6 -6= A - B = 10-2 22-6

1-7 -1-5

OPERATION OF MATRIX

Hal.: 37 Matriks Adaptif

OPERASI MATRIKS

Apa syarat agar dua matriks dapat dijumlahkan?

Jawab:Ordo dua matriks tersebut sama

A = [aij] dan B = [bij] berukuran sama,

A + B didefinisikan: (A + B)ij = (A)ij + (B)ij = aij + bij

Hal.: 38 Matriks Adaptif

OPERATION OF MATRIX

What is the condition so that two matrixes can be added?

Answer:The ordo of the two matrixes are the same

A = [aij] dan B = [bij] have the same size,

A + B is defined: (A + B)ij = (A)ij + (B)ij = aij + bij

Hal.: 39 Matriks Adaptif

Jumlah dua matriks

5 6 1

7 2 3C = 25 30 5

35 10 15D =

C + D = ? ? ?

? ? ?

1 4 -9 3 7 0 5 9 -13

K = 7 3 1-2 4 -5 9 -4 3

L =

K + L =

? ? ?

? ? ?

? ? ?

D + C =

L + K =

Apa kesimpulanmu? Apakah jumlahan matriks bersifat komutatif?

OPERASI MATRIKS

Hal.: 40 Matriks Adaptif

The quantity of two matrixes

5 6 1

7 2 3C = 25 30 5

35 10 15D =

C + D = ? ? ?

? ? ?

1 4 -9 3 7 0 5 9 -13

K = 7 3 1-2 4 -5 9 -4 3

L =

K + L =

? ? ?

? ? ?

? ? ?

D + C =

L + K =

What is your conclusion? Is the addition of matrixes commutative?

OPERATION OF MATRIX

Hal.: 41 Matriks Adaptif

Soal:

C + D =… C + E = … A + B = …

3 -8 0

4 7 2

-1 8 4C = D =

3 7 2

5 2 6

-1 8 4E =

2 7 2

5 2 6

0 0 0

0 0 0 A =

0 0 0

0 0 0B =

6 -1 2

9 9 8

-2 16 8C +D = Feedback:

OPERASI MATRIKS

Hal.: 42 Matriks Adaptif

Exercise:

C + D =… C + E = … A + B = …

3 -8 0

4 7 2

-1 8 4C = D =

3 7 2

5 2 6

-1 8 4E =

2 7 2

5 2 6

0 0 0

0 0 0 A =

0 0 0

0 0 0B =

6 -1 2

9 9 8

-2 16 8C +D = Feedback:

OPERATION OF MATRIX

Hal.: 43 Matriks Adaptif

Hasil kali skalar dengan matriks

5 6 1

7 2 3A = 5A = =

250 300 50

350 100 150H = H =

Diberikan matriks A = [aij] dan skalar c, perkalian skalar cA mempunyai entri-entri sebagai berikut:

(cA)ij = c.(A)ij = caij

Apa hubungan H dengan A?

5x5

5x5

5x6

5x2

5x1

5x3

25

35

30

10

5

15

Catatan: Pada himpunan Mmxn, perkalian matriks dengan skalar bersifat tertutup (menghasilkan matriks dengan ordo yang sama)

50A

OPERASI MATRIKS

Hal.: 44 Matriks Adaptif

The multiplication result of scalar matrix

5 6 1

7 2 3A = 5A = =

250 300 50

350 100 150H = H =

Given matrix A = [aij] aand scalar c, the multiplication of scalar cA have the following entries:

(cA)ij = c.(A)ij = caij

What is the relation between H and A?

5x5

5x5

5x6

5x2

5x1

5x3

25

35

30

10

5

15

Note: In the set of Mmxn, the matrix multiplication with scalar have closed properties (it will have matrix with the same orrdo)

50A

OPERATION OF MATRIX

Hal.: 45 Matriks Adaptif

OPERASI MATRIKS K 3 x 3

1 4 -9 3 7 0 5 9 -13

K =

5 20 -4515 35 0

25 45 -655K =

4 16 -36 12 28 0

20 36 -524K =

Hal.: 46 Matriks Adaptif

OPERATION OF MATRIX K 3 x 3

1 4 -9 3 7 0 5 9 -13

K =

5 20 -4515 35 0

25 45 -655K =

4 16 -36 12 28 0

20 36 -524K =

Hal.: 47 Matriks Adaptif

OPERASI MATRIKS

Diketahui bahwa cA adalah matriks nol. Apa kesimpulan Anda tentang A dan c?

0 0 0

0 0 0 A = A =

2 7 2

5 2 6c = 0c = 7

cA = 0*2 0*7 0*2

0*5 0*2 0*6

0 0 0

0 0 0 = cA =

7*0 7*0 7*0

7*0 7*0 7*0

Kasus 1: c = 0 dan A matriks sembarang. Kasus 2: A matriks nol dan c bisa berapa saja.

Contoh:

kesimpulan

Hal.: 48 Matriks Adaptif

OPERATION OF MATRIX

Known that cA is zero matrix. What is your conclusion about A and c?

0 0 0

0 0 0 A = A =

2 7 2

5 2 6c = 0c = 7

cA = 0*2 0*7 0*2

0*5 0*2 0*6

0 0 0

0 0 0 = cA =

7*0 7*0 7*0

7*0 7*0 7*0

Case 1: c = 0 and A is any matrixCase 2: A is zero matrix and c can be any number

Example:

Conclusion

Hal.: 49 Matriks Adaptif

OPERASI MATRIKS

Definisi: Jika A = [aij] berukuran m x r , dan B = [bij] berukuran r x n,

maka matriks hasil kali A dan B, yaitu C = AB mempunyai elemen-elemen yang didefinisikan sebagai berikut:

∑ aikbkj = ai1b1j +ai2b2j+………airbrj

k = 1

(C)ij = (AB)ij =

2 3 4 5

8 -7 9 -4

1 -5 7 -8

A = 1 2

7 -6

4 -9

B = Tentukan AB dan BA

A B ABm x r r x n m x n

• Syarat:

r

Perkalian matriks dengan matriks

Hal.: 50 Matriks Adaptif

OPERATION OF MATRIX

Definition: If A = [aij] have size m x r , and B = [bij] have size r x n,

then the matrix which is from the multiplication result between A and B, yaitu is C = AB has elements that defined as follows:

∑ aikbkj = ai1b1j +ai2b2j+………airbrj

k = 1

(C)ij = (AB)ij =

2 3 4 5

8 -7 9 -4

1 -5 7 -8

A = 1 2

7 -6

4 -9

B = Define AB and BA

A B ABm x r r x n m x n

• Condition:

r

Multiplication between matrix

Hal.: 51 Matriks Adaptif

Perkalian matriks dengan matriks

2 3 4 5

8 -7 9 -4

1 -5 7 -8

A =

1 2

7 -6

4 -9

11 3

B =

A B = 2.1 +3.7+4.4+5.11 -35

-49 -35

-94 -55

94 -35

-49 -35

-94 -55

=

OPERASI MATRIKS

=

Contoh :

BA tidak didefinisikan

Hal.: 52 Matriks Adaptif

The multiplication between matrixes

2 3 4 5

8 -7 9 -4

1 -5 7 -8

A =

1 2

7 -6

4 -9

11 3

B =

A B = 2.1 +3.7+4.4+5.11 -35

-49 -35

-94 -55

94 -35

-49 -35

-94 -55

=

OPERATION OF MATRIX

=

Example:

BA is not define

Hal.: 53 Matriks Adaptif

OPERASI MATRIKS1. Diberikan A dan B, AB dan BA terdefinisi. Apa kesimpulanmu?

2. AB = O matriks nol, apakah salah satu dari A atau B pasti matriks nol?

2 32 3A = 3 -3

-2 2B = 0 00 0AB =

B An x k m x n

m = k

ABmxm ABnxn

AB dan BA matriks persegi

AB matriks nol, belum tentu A atau B matriks nol

A Bn x km x n

Hal.: 54 Matriks Adaptif

OPERATION OF MATRIX1. Given A and B, AB and BA is defined. What is your conclusion?

2. AB = O is zero matrix, is one of (A or B) is zero matrix?

2 32 3A = 3 -3

-2 2B = 0 00 0AB =

B An x k m x n

m = k

ABmxm ABnxn

AB and BA square matricx

AB is zero matrix. Matrix A and B is not certain zero matrix

A Bn x km x n

Hal.: 55 Matriks Adaptif

Tentukan hasil kalinya jika terdefinisi.

• A B = ??• AC = ??• BD = ??• CD = ??• DB = ??

OPERASI MATRIKS

2 3 4 5 4 7 9 0 2 3 5 6

A = 1 2-9 0 8 0 5 6

B =

7 -11 43 5 -6

C = 1 8 9 5 6 2 5 6 -9 0 0 -4 7 8 9

D =

Contoh 1:

Hal.: 56 Matriks Adaptif

Define the multiplication result if it defined:

• A B = ??• AC = ??• BD = ??• CD = ??• DB = ??

OPERATION OF MATRIX

2 3 4 5 4 7 9 0 2 3 5 6

A = 1 2-9 0 8 0 5 6

B =

7 -11 43 5 -6

C = 1 8 9 5 6 2 5 6 -9 0 0 -4 7 8 9

D =

Example 1:

Hal.: 57 Matriks Adaptif

OPERASI MATRIKS

Contoh 2:2 31 2A =

A2 = 2 31 2

2 31 2

A3 = A x A2 = 2 31 2

2 31 2

2 31 2

A0 = IAn =

n faktor

An+m = An Am

A A A …A

Hal.: 58 Matriks Adaptif

OPERATION OF MATRIX

Example 2:2 31 2A =

A2 = 2 31 2

2 31 2

A3 = A x A2 = 2 31 2

2 31 2

2 31 2

A0 = IAn =

n factor

An+m = An Am

A A A …A

Hal.: 59 Matriks Adaptif

DETERMINAN DAN INVERS

Kompetensi Dasar:Menentukan determinan dan invers

Indikator :1. Matriks ditentukan determinannya2. Matriks ditentukan inversnya

Hal.: 60 Matriks Adaptif

DETERMINANT AND INVERSE

Basic Competence:Define the determinant and inverseIndicator :1. Matrix is defined by its determinant2. Matrix is defined by its inverse

Hal.: 61 Matriks Adaptif

DETERMINAN DAN INVERS

Determinan Matriks ordo 2 x 2

Nilai determinan suatu matriks ordo 2 x 2 adalah hasil kali elemen-elemen diagonal utama dikurangi hasil kali elemen pada diagonal kedua.

Misalkan diketahui matriks A berordo 2 x 2, A =

Determinan A adalah

det A =

dcba

dcba

= ad - bc

Hal.: 62 Matriks Adaptif

DETERMINANT AND INVERSE

Determinant Matrix ordo 2 x 2Determinant value of a matrix ordo 2 x 2 is the multiplication result of the main diagonal elements and subtract by the multiplication result of the second diagonal.

For example, known matrix A ordo 2 x 2, A =

Determinant A is

det A =

dcba

dcba

= ad - bc

Hal.: 63 Matriks Adaptif

Contoh: Invers matriks 2x2

3 2

4 1A =

I=

1 -23.1-4.2 3.1-4.2

3-43.1-4.2 3.1-4.2

=A-1

1 25 5

345 5

DETERMINAN DAN INVERS

Hal.: 64 Matriks Adaptif

Example: Matrix inverse 2x2

3 2

4 1A =

I=

1 -23.1-4.2 3.1-4.2

3-43.1-4.2 3.1-4.2

=A-1

1 25 5

345 5

DETERMINANT AND INVERSE

Hal.: 65 Matriks Adaptif

DETERMINANT DAN INVERSE

1. Kapan matriks TIDAK mempunyai invers? a bc d

2. Tentukan invers matriks berikut ini

1 0

0 1d.

5 1

1 2a.

0 1

0 2b.

0 0

4 1c.

1 0

0 1d.

2/3 -1/5

-1/5 5/3a.

ad-bc = 0

b. tidak mempunyai invers

c. tidak mempunyai invers

Contoh :

Hal.: 66 Matriks Adaptif

DETERMINANT AND INVERSE

1. When matrix Doesn’t have inverse? a bc d

2. Define the following matrix inverse

1 0

0 1d.

5 1

1 2a.

0 1

0 2b.

0 0

4 1c.

1 0

0 1d.

2/3 -1/5

-1/5 5/3a.

ad-bc = 0

b. Doesn’t have inverse

c. Doesn’t have inverse

Example :

Hal.: 67 Matriks Adaptif

DETERMINAN DAN INVERS

B adalah invers dari matriks A, jika AB = BA = I matriks identitas, ditulis B = A-1

dcba

A IA-1A-1 A= =

Jika A = , maka

acbd

bcadA 11

0 bcadAdengan

Hal.: 68 Matriks Adaptif

DETERMINANT AND INVERSE

B is inverse of matrix A, if AB = BA = I matrix identities, it is written B = A-1

dcba

A IA-1A-1 A= =

If A = , then

acbd

bcadA 11

0 bcadAwith

Hal.: 69 Matriks Adaptif

DETERMINAN DAN INVERS

Contoh 1 :Tentukan invers dari matriks

27517

5.717.211

acbd

AA

42

10517752

danBA

Jawab :

27517

det B = (-5) . (-4) – (-2) . (-10) = 20 – 20 = 0 , sehingga matriks B

tidak memiliki invers

Hal.: 70 Matriks Adaptif

DETERMINANT AND INVERSE

Example 1 :Defined the inverse of matrix

27517

5.717.211

acbd

AA

42

10517752

danBA

Answer :

27517

det B = (-5) . (-4) – (-2) . (-10) = 20 – 20 = 0 , So, matrix B doesn’t

have inverse

Hal.: 71 Matriks Adaptif

DETERMINAN DAN INVERS

1 0 0

0 1 0

0 0 1

4 2

2 2

½ -½

-½ 1

4 2 1

2 2 1

3 3 1

½ -½ 1

-½ -½ 1

0 3 -2

1 0

0 1

Contoh 2 :

4 2

2 2

½ -½

-½ 1= =

A A-1 A-1 A I

4 2 1

2 2 1

3 3 1

½ -½ 1

-½ -½ 1

0 3 -2

= =

B B-1 B-1 B I

Diketahui matriks

Tunjukkan bahwa A.A-1 = A-1.A = I dan B.B-1 = B-1. B = I

133122124

2224

danBA

Hal.: 72 Matriks Adaptif

DETERMINANT AND INVERSE

1 0 0

0 1 0

0 0 1

4 2

2 2

½ -½

-½ 1

4 2 1

2 2 1

3 3 1

½ -½ 1

-½ -½ 1

0 3 -2

1 0

0 1

Example 2 :

4 2

2 2

½ -½

-½ 1= =

A A-1 A-1 A I

4 2 1

2 2 1

3 3 1

½ -½ 1

-½ -½ 1

0 3 -2

= =

B B-1 B-1 B I

Known matrix

Show that A.A-1 = A-1.A = I and B.B-1 = B-1. B = I

133122124

2224

danBA

Hal.: 73 Matriks Adaptif

DETERMINAN DAN INVERS

Matriks ordo 3 x 3

.

ihgfedcba

MisalkanA

Determinan Matriks Ordo 3 x 3

Dengan aturan Sarrus, determinan A adalah sebagai berikut.

hgedba

ihgfedcba

A

_ _ _ + + +bdiafhcegcdhbfgaei

)()( bdiafhcegcdhbfgaei

Hal.: 74 Matriks Adaptif

DETERMINANT AND INVERSE

Matrix ordo 3 x 3

.

ihgfedcba

exampleA

Matrix Determinant Ordo 3 x 3

With Sarrus rule, determinant A is as follows

hgedba

ihgfedcba

A

_ _ _ + + +bdiafhcegcdhbfgaei

)()( bdiafhcegcdhbfgaei

Hal.: 75 Matriks Adaptif

DETERMINAN DAN INVERS

Sistem Persamaan Linear Dua Variabel dengan Menggunakan Matriks

Misal SPL 111 cybxa

222 cybxa

Persamaan tersebut dapat di ubah menjadi bentuk matriks

berikut

2

1

22

11

cc

yx

baba

Hal.: 76 Matriks Adaptif

DETERMINANT AND INVERSE

The equation of linear with two variable using matrix

For example SPL 111 cybxa

222 cybxa

The equation can be changed into the following matrix

2

1

22

11

cc

yx

baba

Hal.: 77 Matriks Adaptif

DETERMINAN DAN INVERS

Misalkan ,,2

1

22

11

CC

danByx

Pbaba

A maka dapat ditulis

2

1

22

11

cc

yx

baba

BAP

BAP 1

Hal.: 78 Matriks Adaptif

DETERMINANT AND INVERSE

Example ,,2

1

22

11

CC

andByx

Pbaba

A Then can be write

as

2

1

22

11

cc

yx

baba

BAP

BAP 1

Hal.: 79 Matriks Adaptif

DETERMINAN DAN INVERS

Contoh :

1632 yxTentukan nilai x dan y yang memenuhi sistem persamaan linear

134 yx

134 yxJawab :

Sistem persamaan 1632 yx

Jika dibuat dalam bentuk matriks menjadi

1316

4132

yx

Hal.: 80 Matriks Adaptif

DETERMINANT AND INVERSE

Example:

1632 yxDefine the value of x and y that fulfill the equation of linear system

134 yx

134 yxanswer :

Equation system1632 yx

If in matrix

1316

4132

yx

Hal.: 81 Matriks Adaptif

DETERMINAN DAN INVERS

Perkalian matriks berbentuk AP = B dengan

1316

,41

32danB

yx

PA

2134

51

2134

3.14.211A

BAP

BAP 1

1316

2134

51

yx

25

1025

51

26163964

51

Jadi nilai x = 5 dan y = 2

Hal.: 82 Matriks Adaptif

DETERMINANT AND INVERSE

The matrix multiplication in the form of AP = B with

1316

,41

32danB

yx

PA

2134

51

2134

3.14.211A

BAP

BAP 1

1316

2134

51

yx

25

1025

51

26163964

51

So, the value of x = 5 and y = 2

Hal.: 83 Matriks Adaptif

DETERMINAN DAN INVERS

Penyelesaian sistem persamaan linear dua variabel dengan menggunakan determinan atau aturan Cramer.

cbyax Misal SPL

rqypx

Maka dengan aturan Cramer, diperoleh

,

qpbaqrbc

x

qpbarpca

y dan

Hal.: 84 Matriks Adaptif

DETERMINANT AND INVERSE

The solution of linear equation system with two variables using determinant or Cramer rule

cbyax For example

SPLrqypx

Then, with Cramer rule, we get

,

qpbaqrbc

x

qpbarpca

y dan

Hal.: 85 Matriks Adaptif

DETERMINAN DAN INVERS

Contoh :Gunakan aturan Cramer untuk menentukan himpunan penyelesaian sistem persamaan linear

543 yx

42 yx

11111

)4.(21.3)4.(41).5(

12431445

x

Jawab :

Dengan aturan Cramer diperoleh

21122

)4.(21.3)5.(24.3

1243

4253

y

Jadi, himpunan penyelesaiannya adalah {(1,2)}.

Hal.: 86 Matriks Adaptif

DETERMINANT AND INVERSE

Example :Use the Cramer rule to define the solution set of linear equation system

543 yx

42 yx

11111

)4.(21.3)4.(41).5(

12431445

x

answer :

With cramer Rule, we get

21122

)4.(21.3)5.(24.3

1243

4253

y

So, the solution set is {(1,2)}.

Hal.: 87 Matriks Adaptif

DETERMINAN DAN INVERS

Menyelesaikan Sistem Persamaan Linear Tiga Variabel dengan menggunakan Matriks

SPL dalam bentuk:

Dapat disajikan dalam bentuk persamaan matriks:

a11x1 + a12x2 + a13x3 +….. ..a1nxn = b1

a21x1 + a22x2 + a23x3 +…….a2nxn = b2

am1x1 + am2x2 + am3x3 + ……amnxn = bm

a11 a12……...a1n

a21 a22 ……..a2n : : : am1 am2…… amn

x1

x2

:xn

= b1

b2

:bn

A: matriks koefisien

Ax = bx b

Hal.: 88 Matriks Adaptif

DETERMINANT AND INVERSE

Finishing the equation of linear system with three variables using matrix

SPL in the form of:

It can be written in the form of matrix equation:

a11x1 + a12x2 + a13x3 +….. ..a1nxn = b1

a21x1 + a22x2 + a23x3 +…….a2nxn = b2

am1x1 + am2x2 + am3x3 + ……amnxn = bm

a11 a12……...a1n

a21 a22 ……..a2n : : : am1 am2…… amn

x1

x2

:xn

= b1

b2

:bn

A: matrix coefficient Ax = b

x b

Hal.: 89 Matriks Adaptif

DETERMINAN DAN INVERS

x1 + 2x2 + x3 = 6

-x2 + x3 = 1

4x1 + 2x2 + x3 = 4

SPL

1 2 1

0 -1 1

4 2 1

x1

x2

x3

=6

1

4

1.x1 +2.x2 + 1.x3

0.x1 + -1.x2 + 1.x3

4.x1 +2.x2 + 1.x3

=6

1

4

Dapat disajikan dalam bentuk matriks sebagai berikut

Contoh :

Hal.: 90 Matriks Adaptif

DETERMINANT AND INVERSE

x1 + 2x2 + x3 = 6

-x2 + x3 = 1

4x1 + 2x2 + x3 = 4

SPL

1 2 1

0 -1 1

4 2 1

x1

x2

x3

=6

1

4

1.x1 +2.x2 + 1.x3

0.x1 + -1.x2 + 1.x3

4.x1 +2.x2 + 1.x3

=6

1

4

It can be written in the form of the following matrix

Example :

Hal.: 91 Matriks Adaptif

Perkalian dengan matriks identitas

1 0 00 1 00 0 1

A= 1 2 37 5 6-9 3 -7

A.I = 1 2 37 5 6-9 3 -7

=

1 0 00 1 00 0 1

I.A = =1 2 37 5 6-9 3 -7

1 2 37 5 6-9 3 -7

1 2 37 5 6-9 3 -7

X

DETERMINAN DAN INVERS

X

Hal.: 92 Matriks Adaptif

The multiplication of identity matrix

1 0 00 1 00 0 1

A= 1 2 37 5 6-9 3 -7

A.I = 1 2 37 5 6-9 3 -7

=

1 0 00 1 00 0 1

I.A = =1 2 37 5 6-9 3 -7

1 2 37 5 6-9 3 -7

1 2 37 5 6-9 3 -7

X

DETERMINANT AND INVERSE

X

Hal.: 93 Matriks Adaptif

DETERMINAN DAN INVERS

AB = A dan BA = A, apa kesimpulanmu?

1 4 -9 3 7 0 5 9 -13

1 4 -9 3 7 0 5 9 -13

AB = A dan BA = A, maka B = I

(I matriks identitas)

1 0 00 1 0 0 0 1

1 0 00 1 0 0 0 1

=

=

1 4 -9 3 7 0 5 9 -13

1 4 -9 3 7 0 5 9 -13

A AII A= =

Hal.: 94 Matriks Adaptif

DETERMINANT AND INVERSE

AB = A and BA = A, what is your conclusion?

1 4 -9 3 7 0 5 9 -13

1 4 -9 3 7 0 5 9 -13

AB = A and BA = A, then B = I

(I identity matrix )

1 0 00 1 0 0 0 1

1 0 00 1 0 0 0 1

=

=

1 4 -9 3 7 0 5 9 -13

1 4 -9 3 7 0 5 9 -13

A AII A= =

Hal.: 95 Matriks Adaptif

d -bab-cd ab-cd

-c aab-cd ab-cd

DETERMINAN DAN INVERS

4 2

2 2

½ -½

-½ 1

1 0

0 1

d -b

-c a

1

ad - bc

Jika ad –bc = 0 maka A TIDAK mempunyai invers.

=

A IA-1

a b

c d A-1

1 0

0 1=

A-1 = =

Hal.: 96 Matriks Adaptif

d -bab-cd ab-cd

-c aab-cd ab-cd

DETERMINANT AND INVERSE

4 2

2 2

½ -½

-½ 1

1 0

0 1

d -b

-c a

1

ad - bc

If ad –bc = 0 then A doesn’t have inverse

=

A IA-1

a b

c d A-1

1 0

0 1=

A-1 = =

Hal.: 97 Matriks Adaptif

DETERMINAN DAN INVERS

1. Invers dari matriks jika ada adalah tunggal: Jika B = A-1 dan C = A-1, maka B = C

4 2

2 2A =

½ -½

-½ 1A-1

4 2

2 2

1 0

0 1

2. (A-1)-1 = A

?

(A-1)-1

= ½ -½

-½ 1A-1 =

A

Hal.: 98 Matriks Adaptif

DETERMINANT AND INVERSE

1. If there is inverse of matrix is only one: If B = A-1 and C = A-1, then B = C

4 2

2 2A =

½ -½

-½ 1A-1

4 2

2 2

1 0

0 1

2. (A-1)-1 = A

?

(A-1)-1

= ½ -½

-½ 1A-1 =

A

Hal.: 99 Matriks Adaptif

DETERMINAN DAN INVERS

3. Jika A mempunyai invers maka An mempunyai invers dan (An)-1 = (A-1)n, n = 0, 1, 2, 3,…

4 2

2 2A =

4 2

2 2A3 =

4 2

2 2

4 2

2 2

½ -½

-½ 1A-1 =

=104 64

64 40

(A3)-1 = 0.625 -1

-1 1.625

(A-1)3 = 0.625 -1

-1 1.625

½ -½

-½ 1

½ -½

-½ 1

½ -½

-½ 1=

sama

Hal.: 100 Matriks Adaptif

DETERMINANT AND INVERSE

3. If A have inverse then An have inverse and (An)-1 = (A-1)n, n = 0, 1, 2, 3,…

4 2

2 2A =

4 2

2 2A3 =

4 2

2 2

4 2

2 2

½ -½

-½ 1A-1 =

=104 64

64 40

(A3)-1 = 0.625 -1

-1 1.625

(A-1)3 = 0.625 -1

-1 1.625

½ -½

-½ 1

½ -½

-½ 1

½ -½

-½ 1=

The same

with

Hal.: 101 Matriks Adaptif

DETERMINAN DAN INVERS

4. (AB)-1 = B-1 A-1

4 2

2 2A =

3 5

2 2 B = B-1 =

½ 5/4

½ - ¾

(AB)-1 = 16 24

10 14

-1= -0.875 1.5

0.625 -1

A-1 B-1 = ½ 5/4

½ - ¾

½ -½

-½ 1= -0.5 1

0.75 -1.375

B-1 A-1 = ½ 5/4

½ - ¾

½ -½

-½ 1= -0.875 1.5

0.625 -1

Hal.: 102 Matriks Adaptif

DETERMINANT AND INVERSE

4. (AB)-1 = B-1 A-1

4 2

2 2A =

3 5

2 2 B = B-1 =

½ 5/4

½ - ¾

(AB)-1 = 16 24

10 14

-1= -0.875 1.5

0.625 -1

A-1 B-1 = ½ 5/4

½ - ¾

½ -½

-½ 1= -0.5 1

0.75 -1.375

B-1 A-1 = ½ 5/4

½ - ¾

½ -½

-½ 1= -0.875 1.5

0.625 -1