12. matriks.ppt [Read-Only] -...
Transcript of 12. matriks.ppt [Read-Only] -...
slide 0
Maksimisasi utilitas dengan pendapatan
terbatas
� Utility Function
� Budget Constraint
121 2xxxU +=
6024 =+ xx
slide 1
� Lagrangian
602421
=+ xx
[ ]21121 24602 xxxxxL −−++= λ
� Necessary Conditions
042
02460
2
21
=−+=∂
∂
=−−=∂
∂
λ
λ
xL
xxL
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� Tentukan nilai x1 dan x2
02
042
1
1
2
1
=−=∂
∂
=−+=∂
λ
λ
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
=
slide 8
( ) ( )( ) ( )( ) ( )
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
−=−+
−=+−−
λ
λ
xx
xx
042
02460
2
21
=−+=∂
∂
=−−=∂
∂
λ
λ
xx
L
xxL
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020
240
21
21
=−+
−=−+
λ
λ
xx
xx
02
042
1
1
2
1
=−=∂
∂
=−+=∂
λ
λ
xx
L
xx
Matriks
Ax d=
−
−− 60024 1x
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−=
−
−
0
2
201
410 2
1
λ
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
λ
x
Solusi
1x A d
−=
−
−−
−60024
1
1x
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−
−
−=
0
2
201
4102
1
λ
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
slide 18
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 −−−−
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201
410
024
−
−
−−
0
1
2
1
0
4
201
410
024 −−
−
−
−−
Langkah 2: Jumlahkan perkalian
diagonal
Matriks 3x3
24024 −−−−
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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:
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201
410
024
−
−
−−
=A
slide 24
201
410
024
)1(11
11
−
−
−−
−= +C
20
4111
−
−=C
Adjoint
= 322212
312111
.
CCC
CCC
CCC
Aadj
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332313 CCC
Matrix Operations in Excel
slide 26
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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