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2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Week 6
Matrices
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
OverviewYou have already known that matrices used
throughout the discrete mathematics can express the relationships between elements (data/entries) in sets. Frequently, the data is arranged in arrays, that is, sets whose elements are indexed by one or more subscripts.
Technically speaking, matrices will be used in models of communications networks and transportation systems. And there are many algorithms will be developed that use these matrix models.
We shall begin our discussion into two parts, those are less and more than 3-order matrices.
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
Objectives
� Definition of matrix and its components
� The Arithmatic operation of matrix
� Transpose of matrix
� Determinant
� Matrix Inversion
� Multiplying operation between matrix
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
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 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.
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
IndexColumns (n)
Rows (m)Entry
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
Set A = {1,2,3,4,5,6,7}
In matrix becomes
[A] = [ 1 2 3 4 5 6 7 ] � Vector A
In matrix becomes
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
Or in linear equation :
2x + 2y = 16 ……………….(a)
x + 3y = 18 ………………...(b)
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
The Arithmatic operation of matrix
�Addition and Subtraction
�Multiplying with scalar
Addition or Substraction of two or more matrices needs the same index of them.
± ±
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
Multiplying scalar with a matrix basically multiplying a scalar with all entries of matrix
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
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
interchanging the rows and columns of A, AT=[aji].
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
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|
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
The determinants of order 1, 2, and 3 are defined as follows:
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
The following diagram may help you to find the determinant of order 2:
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
The following diagram may help you to find the determinant of order 3:
How about the order of 4, 5, 6,…?
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
Exercises :
Find
(a) A+B;
(b) (b) 3A and -4B
2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
MatricesMatrices
Find the transposition of :