This document discusses the application of matrices in analyzing economic sectors. It begins with an introduction on how relations and matrices from discrete mathematics can be used to analyze inputs and outputs in the economic field. It then covers matrix theory such as identity matrices and types of matrices. It also discusses transaction matrices and technology matrices, which are used to show relationships between sector outputs, inputs, and final demand. An example application is provided to calculate the total outputs required for each sector based on target final demands. The document concludes by listing references used.
3. Pendahuluan
Relasi dan matriks adalah salah satu bagian dari ilmu matematika diskrit yang menarik
untuk dipelajari. Dalam hal ini relasi dan matriks dikembangkan dalam menganalisis
masukan dan keluaran dalam bidang ekonomi.
Masukan dan keluaran yang terbentuk yaitu dari sektor sektor produksi dan konsumen
terdistribusi secara acak yang membentuk hubungan / relasi yang bisa dianalisis.
4. Teori
o Matriks Bujur Sangkar
o Matrik Diagonal
o Matrik Identitas
b. Jenis-jenis Matriks
Matriks merupakan kumpulan angka / bilangan
dalam bentuk segi empat yang terdiri dari baris
dan kolom. Notasi suatu matriks biasanya ditulis
sebagai berikut:
𝑀 =
𝑎11
𝑎21
𝑎12
𝑎22
… 𝑎1𝑗
… 𝑎2𝑗
⋮ ⋮ ⋱ ⋮
𝑎𝑖1 𝑎𝑖2
… 𝑎𝑖𝑗
a. Definisi Matriks
o Matriks Transpos
o Invers Matriks
o Matriks Transaksi
o Matriks teknologi
5. Matriks Trasaksi
Langkah awal dalam analisis masukan-
keluaran adalah menyusun suatu tabel
yang berisi keterang-keterangan tentang
bagaimana keluaran suatu sektor
terdistribusi ke sektor-sektor lain sebagai
masukan dan ke pemakai akhir sebagai
barang konsumsi.
Hubungan terakhir input dan output
disajikan dalam bentuk Sebuah tabel
yang disebut matriks transaksi, seperti
table disamping.
6. Matriks Teknologi
Koefisien teknologi adalah suatu rasio yang menjelaskan
jumlah nilai keluaran sektor i yang diperlukan sebagai
masukan unit di sektor j terhadap jumlah total masukan
pada sektor j.
Matriks Teknologi terbentuk dari sejumlah koefisien
teknologi 𝑎𝑖𝑗 yang terbentuk dari elemen matriks transaksi
7. Sedangkan nilai yang tidak konstan dibentuk dalam
matriks kolom, yaitu permintaan akhir (U) dan keluaran
total (V).
Dari persamaan dan matriks di atas dapat dibuktikan
persamaan yang menunjukkan hubungan kedua peubah
tersebut (U dan V) adalah
8. dan dibentuk perkalian matriks menjadi :
berdasarkan fungsi balikan matriks juga berlaku:
𝑉 = 𝐼 − 𝑀 −1
𝑈
keterangan:
V : Matriks “keluaran total”
U : Matriks “permintaan akhir”
A : Matriks “koefisien teknologi”
I : Matriks Identitas
9. Aplikasi
Masukan
keluaran
P I J PA T
P 20 35 115 30 200
I 100 75 20 80 275
J 30 82 15 190 317
NT 25 90 200 50 365
T 175 282 350 350 1.157
Keterangan
P : Sektor Pertanian
I : Sektor Industri
J : Sektor Jasa
NT : Nilai Tambah
PA : Penilaian Akhir
T : Keluaran Total
10. Pemerintah menargetkan masing – masing :
1. Pertanian : ditingkatkan dari 30 menjadi 250
2. Industri : ditingkatkan dari 80 menjadi 125
3. Jasa : diturunkan dari 190 menjadi 100
14. Setelah melakukan perhitungan tersebut maka masing-masing sektor harus
memasang keluaran total sebesar:
1. Pertanian : ditingkatkan dari 175 menjadi 364,755.
2. Industri : ditingkatkan dari 282 menjadi 402,307.
3. Jasa : diturunkan dari 350 menjadi 164,407.
15. Referensi
Dumairy. (1999). Matematika Terapan untuk Bisnis dan Ekonomi. BPFE, 317.
Rahayu, Y. (2012). IMPLEMENTASI MATRIKS PADA MATEMATIKA BISNIS DAN
EKONOMI. Techno.COM, 74-81.
yapfrine, S. J. (2015). Penerapan Matriks dalam Analisis Sektor Perekonomian
Indonesia. Makalah IF2123 Aljabar Geometri, 1-2.
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Mercury’s name has nothing to do
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Mercury’s name has nothing to do
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Venus has a beautiful
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Venus
Mercury is the closest
planet to the Sun and the
smallest one
Mercury
23. It’s the closest
planet and the
smallest in the
Solar System
Mercury
It has a beautiful
name and is the
second planet
from the Sun
Venus
Despite being
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Mercury is the closest planet
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Despite being red, Mars is
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Mercury is the
closest planet to
the Sun
Mercury
Venus has a
beautiful name,
but it’s terribly hot
Venus
Despite being red,
Mars is actually a
cold place
Mars
It’s the biggest
planet in the Solar
System
Jupiter
Saturn is the
ringed one and a
gas giant
Saturn
31. Mercury is the
closest planet to
the Sun
Mercury
Example
Mars is a cold
place. It’s full of
iron oxide dust
Mars
32. Mars
Curiosities Around the World
Despite being red,
Mars is a cold place
Mercury
Mercury is the
smallest planet
33. Recommendations
Mercury is the
closest planet to
the Sun
Mercury
It’s the biggest
planet in the Solar
System
Jupiter
Saturn is the
ringed one and a
gas giant
Saturn
Despite being red,
Mars is actually a
cold place
Mars
Neptune is the
farthest planet
from the Sun
Neptune
Venus has a
beautiful name, but
it’s terribly hot
Venus
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Mercury is the
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Despite being red,
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Saturn is the
ringed one and a
gas giant
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