Cjr Struktur Aljabar1).docx

Paper of Algebra Structure

CRITICAL JOURNAL REVIEW “ALGEBRA STRUCTURE OF GROUPS IN INVERTIBLE MATRIX SETS”

Lecture : Dr. Izwita Dewi, M.Pd

Arranged by :

Meidy Adelina Lumban Toruan ( 4163312017)

MATHEMATICS DEPARTEMENT FACULTY OF MATHEMATICS AND SCIENCES STATE UNIVERSITY OF MEDAN 2019

PREFACE Thank you, the author of the presence of God Almighty, because thanks to his grace the author can compile and present the Critical Journal Review (CJR) with the title "Study of Algebra Structure of Groups in Invertible Matrix Sets". This CJR is made to fulfill the assignments " Algebraic Structures ". Do not forget the author thanked the various parties who have provided encouragement and motivation. The author realizes that in the preparation of the CJR there are still many shortcomings and far from perfection. Therefore, the author expects constructive criticism and suggestions to perfect this task and can be a reference in arranging further assignments. The author also apologizes if in writing this task there are typing errors and errors that confuse the reader in understanding the author's intent.

Medan, March 12th 2019

Author

CHAPTER I INTRODUCTION

Matrix is a collection of elements arranged according to rows and columns so that they are rectangular, whose length and width are indicated by the number of columns and rows. Matrices can be grouped as a set based on a certain character and certain types of entries. In general, the Mm notation, n (k) describes a m × n sized matrix whose entry is in the field k. Mm, n (k) is a dimensionless vector space with a scalar binary sum and multiplication operation, so that it can further be investigated for the properties that apply to the set of matrices based on character, the field with matrix entries and binary operations that apply to the matrix. From the description above, the validity of group axioms and subgroups in a set of matrices that have inverse (invertible) will be examined. This scientific work aims to prove the validity of group requirements in an invertible matrix set and examine the shape of subsets which are also groups of matrix multiplication operations which are also subgroups of invertible matrices.

1.1 Background

Adapun yang menjadi latar belakang dalam penulisan makalah ini adalah untuk memenuhi persyaratan dari tugas kuliah sekaligus menambah wawasan .Critical Journal Review bagi mahasiswa adalah tugas wajib yang harus dikerjakan dimana tugas CJR ini adalah tugas yang telah diberikan oleh Dosen kepada mahasiswa disetiap smesternya. Dimana tugas CJR tersebut telah tercantum di kurikulum baru berbasis KKNI yang baru-baru ini digunakan di Universitas Negeri Medan, CJR akan menjadi bagian penting di setiap semester perkuliahan. Yang menjadi latar belakang penulis memilih judul ini yaitu, terlihat dari judul yang sangat terkait erat dengan subsub pokok pembahasan sebuah materi yang terdapat pada Mata kuliah struktur aljabar. Dimana pada jurnal ini memiliki judul membawa . Dalam sebuah buku struktur aljabar terdapat sebuah pokok pembahasan mengenai matriks. Inilah yang melatar belakangi penulis memilih jurnal tersebut.

1.2 Purpose

a. To find out the important role of the journal. b. To find out the method used in the journal. c. To find out the results of the research journal. d. To find out the assessment of the journal.

CHAPTER II

2.1. JOURNAL IDENTITY A. Journal I

Title

: Kajian Struktur Aljabar Grup Pada Himpunan Matriks Yang Invertibel

Author

: Novi Rustiana Dewi, Ning Eliyati, dan Oktavianus Hasiholan Marbun

Volume

: Volume 14 Nomer 1(A) 14101 Jurnal Penelitian Sains

Page

: 3 Pages

Year

: January 2014

B. Journal II Title

: K- Aljabar

Author

: Iswati dan Suryoto

Volume

: Vol. 13, No.1 Jurnal Matematika

Page

: 20 - 33

Tahun

: April 2010

C. Journal III Judul

: Invers Matriks Tergenalisasi Atas Aljabar Maxplus

Penulis

: Mustofa

Volume dan halaman : vol.7, No.1,20-30 Jurnal Matematika dan Komputer Tahun

: 30 April 2004

ISNN

: 1410 - 8518

2.2. Journal Review Journal I According to Bakerjika given Mm, n (k) a set of m × n sized matrices whose entries are in Field k, then denoted entries (i, j) of a matrix A measuring m × n with Aij or aij and...

Furthermore, special notation is used: Mn (k) = Mn, n (k), kn = Mn, l (k) Mm, n (k) is a vector-k space with a scalar and multiplication matrix operation. The null vector is the zero Om matrix, n which is usually denoted by O only. The following are given some definitions and theorems related to group studies in an invertible set of matrices, beginning with some definitions and properties of square matrices. Definition 1 The square matrix (squarematrix) is matirks whose same number of rows and columns are denoted by the An matrix, n = An. Teorema 1 f A and B are square matrices with the same order, then det (AB) = det (A) det (B) Teorema 2 f A is a matrix that has an inverse then A−1 = 1 adj(A) [1].

Definisi 2 Identity matrices are also called unit matrices, which are denoted by "I", are cage budget matrices where all the main diagonal elements are equal to 1, and all other elements are equal to 0. The following are given definitions of group algebraic structures and some group and subgroup properties. Definisi 3 “G is given a non-empty set that is equipped with a binary “ * ”. The set G is said to be a group if it fulfills the following axioms:

(i)

Closed that is for every a, b ∈G then a * b ∈ G.

(iii)

ssociative for every a, b, c ∈ G maka (a*b) *c = a*(b * c)

(iv)

element e ∈ G.

(ii)

There is e e ∈G such that for each a ∈ pplies e * a = a * e = a (there is a neutral

For every a ∈ tt there is a single inverse a−1 ∈ tt so that a ∗ a−1 = a−1 ∗ a = e

Definisi 4 If a subset of H of a group G is closed under a binary operation on G and if H itself is a group, then H is a subgroup of G. Then the notation H * G or G * H states that H is a subgroup of G, and H
Journal II

1. From a K-algebra can be formed a set of parts that have the properties of K-algebra against the same binary operation called K-sub algebra 2. As in the group with the concept of group homomorphism, there is also a homomorphism concept called homomorphism in the K- algebra. 3. The properties that apply to the group, will also apply to K-algebra. Journal 3

If the A matrix is over the field, then there must be a single matrix B that satisfies the nature of ABA = A. The B matrix that satisfies this property is called the generalized inverse matrix A. In maxplus algebra, there is no guarantee that each matrix has generalized inverse. If A has a generalized inverse, then A is said to be regular. CHAPTER III CONCLUSION The results of the discussion can be concluded as follows: 1. A set of matrices invertible GLn (k) with binary multiplication operations is a group 2. A set of SLn (k) from GLn (k) which is a set of matrices whose determinant 1 is a subgroup of GLn (k). K-algebra is an algebra structure built on a group so that characters of a group will apply also at K-algebra. If at group there is subgroup and homomorfism group, hence at Kalgebra there is K-subalgebra and K-homomorfism. By using characters of group, will be proved characters applied at K-algebra.