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TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
Week 6
Matrices
MatricesMatrices
OverviewYou have already known that matrices used
throughout the discrete mathematics can express therelationships between elements (data/entries) in sets.Frequently, the data is arranged in arrays, that is, setswhose elements are indexed by one or more subscripts.
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
whose elements are indexed by one or more subscripts.
Technically speaking, matrices will be used inmodels of communications networks and transportationsystems. And there are many algorithms will be developedthat use these matrix models.
We shall begin our discussion into two parts, thoseare less and more than 3-order matrices.
MatricesMatrices
Objectives
� Definition of matrix and its components
� The Arithmatic operation of matrix
� Transpose of matrix
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
� Transpose of matrix
� Determinant
� Matrix Inversion
� Multiplying operation between matrix
MatricesMatrices
Definition of matrix and its components
A matrix is a rectangular array of data/entries
A matrix with m rows and n columns is called an m x n matrix (read: m by n matrix). A matrix with the same number of rows as columns is
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
m by n matrix). A matrix with the same number of rows as columns is called square matrix.
Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal.
A one-dimensional array of data/entries is called a vector.
MatricesMatricesRows (m)
Entry
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.EngIndex
Columns (n)
MatricesMatrices
Set A = {1,2,3,4,5,6,7}
In matrix becomes
[A] = [ 1 2 3 4 5 6 7 ] � Vector A
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
In matrix becomes
MatricesMatrices
Or in linear equation :
2x + 2y = 16 ……………….(a)
x + 3y = 18 ………………...(b)
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
MatricesMatrices
The Arithmatic operation of matrix
�Addition and Subtraction
�Multiplying with scalar
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
Addition or Substraction of two or more matrices needs the same index of them.
± ±
MatricesMatrices
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
MatricesMatrices
Multiplying scalar with a matrix basically multiplying a scalar with all entries of matrix
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
MatricesMatrices
Transpose of Matrix
Basically, it just changes rows into columns
Let A = [aij] be an m x n matrix. The transpose of
A, denoted by AT, is the n x m matrix obtained by
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
A, denoted by AT, is the n x m matrix obtained by
interchanging the rows and columns of A, AT=[aji].
MatricesMatrices
Determinant
To each n-square matrix A = [aij], we assign a specific number called the determinant of A and denoted by det [A] or |A|
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
A and denoted by det [A] or |A|
MatricesMatrices
The determinants of order 1, 2, and 3 are defined as follows:
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
MatricesMatrices
The following diagram may help you to find the determinant of order 2:
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
MatricesMatrices
The following diagram may help you to find the determinant of order 3:
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
How about the order of 4, 5, 6,…?
MatricesMatrices
Examples:
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
MatricesMatrices
Exercises :
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
Find
(a) A+B;
(b) (b) 3A and -4B
MatricesMatrices
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
MatricesMatrices
Find the transposition of :
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
MatricesMatrices
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
MatricesMatrices
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng