TIF 21101

APPLIED MATH 1

(MATEMATIKA TERAPAN 1)

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

Week 6

Matrices

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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.

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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

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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

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.

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Entry

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.EngIndex

Columns (n)

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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

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

In matrix becomes

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Or in linear equation :

2x + 2y = 16 ……………….(a)

x + 3y = 18 ………………...(b)

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

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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.

± ±

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Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

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

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

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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

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].

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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|

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

A and denoted by det [A] or |A|

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

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

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

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

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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,…?

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Examples:

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

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Exercises :

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

Find

(a) A+B;

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

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Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

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Find the transposition of :

Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

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Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

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Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng