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Maksimisasi utilitas dengan pendapatan

terbatas

Utility Function

Budget Constraint

121 2xxxU +=

6024 =+ xx

slide 1

Lagrangian

602421

=+ xx

[ ]21121 24602 xxxxxL ++=

Necessary Conditions

042

02460

2

21

=+=

==

xL

xxL

slide 2

Tentukan nilai x1 dan x2

02

042

1

1

2

1

==

=+=

xx

L

xx

Bentuk Umum

11 1 12 2 1 1

21 1 22 2 2 2

...

...

............................................

n n

n n

a x a x a x d

a x a x a x d

+ + + =

+ + + =

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1 1 2 2

............................................

...m m mn n ma x a x a x d+ + + =

Ukuran matriks

Matrix [A] akan disebut berukuran mxn jika

mempunyai m baris dan n kolom

Lambangnya adalah [A]mxn

Elemennya disimbolkan dengan a , dimana i

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Elemennya disimbolkan dengan aij, dimana i

merupakan urutan baris dan j urutan kolom.

CHAPTER 10 Aggregate Demand I

Matriks

Ax d=

11 1na a

A

=

K

M O M

1

2

x

xx

=

1

2

d

dd

= M

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1m mna a L

n

x

x

=

M

m

d

d

=

M

mxn ijA a = 1, 2,...,

1,2,...,

i m

j n

=

=

Solusi

Ax d=

1x A d

=

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x A d=

Perkalian Matriks

mxn nxo mxoA B C=

[m x n] dan [n x o]syarat

hasil [m x o]

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hasil [m x o]

Contoh

1 3

28

40

A

=

5

9b

=

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( ) ( )( ) ( )( ) ( )

1 5 3 9 32

2 5 8 9 82

4 5 0 9 20

C

+

= + = +

Contoh

A =

2 3

1 1

1 0

and B =

1 1 1

1 0 2

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1 0 [3 x 2] [2 x 3]

A and B can be multiplied

=

=+=+=+

=+=+=+

=+=+=+

=

1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

[3 x 3]

A =

2 3

1 1

1 0

and B =

1 1 1

1 0 2

Contoh

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=

=+=+=+

=+=+=+

=+=+=+

=

1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

1 0 [3 x 2] [2 x 3]

[3 x 3]

Result is 3 x 3

Operasi matriks

Perkalian skalar

3 1 21 77

0 5 0 35

=

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0 5 0 35

11 12 11 12

21 22 21 22

a a ca cac

a a ca ca

=

Keunikan matriks

AB BA

2 4 2 4

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2 4

1 2A

=

2 4

1 2B

=

240

60024

21

21

=+

=+

xx

xx

042

02460

2

21

=+=

==

xx

L

xxL

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020

240

21

21

=+

=+

xx

xx

02

042

1

1

2

1

==

=+=

xx

L

xx

Matriks

Ax d=

60024 1x

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=

0

2

201

410 2

1

x

Operasi matriks

mxn nxo mxoA B C=

33xA =

=

2

60

410

024 1

x

x

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13

13

xC

xB

=

=

=

0

2

201

410 2

x

Solusi

1x A d

=

60024

1

1x

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=

0

2

201

4102

1

x

Inverse

1 1.

det .A adj A

A

=

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det .A

Determinan

Hanya matriks yang bujursangkar

(baris = kolom) yang mempunyai

determinan

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Matriks 2x2

11 12

21 22

a aA

a a

=

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11 22 12 21det A a a a a=

Langkah 1: Tambahkan 2 kolom

yang pertama

Matriks 3x3

024 24024

slide 20

201

410

024

0

1

2

1

0

4

201

410

024

Langkah 2: Jumlahkan perkalian

diagonal

Matriks 3x3

24024

slide 21

0

1

2

1

0

4

201

410

024

[(-4x1x-2)+(-2x-4x1)+(0x0x0)]-[(1x1x0)+(0x-4x-4)+(-2x0x-2)]

Minors of a Matrix Determinant

A minor Mi,j is a reduced determinant found

by omitting the ith row and jth column of a

larger determinant. For example:

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The Cofactor of Determinants

A cofactor Ci,j is a minor Mi,j augmented with a

sign rule for the particular purpose of solving

matrix determinants. Cofactors are defined as

follows:

slide 23

201

410

024

=A

slide 24

201

410

024

)1(11

11

= +C20

4111

=C

Adjoint

= 322212

312111

.

CCC

CCC

CCC

Aadj

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332313 CCC

Matrix Operations in Excel

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Select the

cells in

which the

answer

will

appear

Matrix Multiplication in Excel

1) Enter

=mmult(

2) Select the

cells of the

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cells of the

first matrix

3) Enter comma

,

4) Select the

cells of the

second matrix

5) Enter )

Matrix Multiplication in Excel

Enter these

three

key

strokes

at the

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at the

same

time:

control

shift

enter

Matrix Inversion in Excel

Follow the same procedure

Select cells in which answer is to be displayed

Enter the formula: =minverse(

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Select the cells containing the matrix to be

inverted

Close parenthesis type )

Press three keys: Control, shift, enter

matrix.xlsx

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