Matematika terapan week 6

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

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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

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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

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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

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IndexColumns (n)

Rows (m)Entry

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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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

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Or in linear equation :

2x + 2y = 16 ……………….(a)

x + 3y = 18 ………………...(b)

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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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

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2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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Multiplying scalar with a matrix basically multiplying a scalar with all entries of matrix

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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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

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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

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The determinants of order 1, 2, and 3 are defined as follows:

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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The following diagram may help you to find the determinant of order 2:

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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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

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Examples:

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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Exercises :

Find

(a) A+B;

(b) (b) 3A and -4B

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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Find the transposition of :

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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