Matrix and its dimension Types of matrix:TYPICAL APPLICATIONS

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MTH001 Elementary Mathematics

LECTURE # 19

Matrix and its dimension

Types of matrix

OBJECTIVES

The objectives of the lecture are to learn about:

Matrices

QUESTIONS

Every student wonders why he or she should study matrices. There are manty

important questions:

Where can we use Matrices?

Typical applications?

What is a Matrix?

What are Matrix operations?

Excel Matrix Functions?

There are many applications of matrices in business and industry especially

where large amounts of data are processed daily.

TYPICAL APPLICATIONS

Practical questions in modern business and economic management can be

answered with the help of matrix representation in:

Econometrics

Network Analysis

Decision Networks

Optimization

Linear Programming

Analysis of data

Computer graphics

WHAT IS A MATRIX?

A Matrix is a rectangular array of numbers. The plural of matrix is matrices.

Matrices are usually represented with capital letters such as Matrix A, B, C.

For example

The numbers in a matrix are often arranged in a meaningful way. For example,

the

order for school clothing in September is illustrated in the table, as well as in the

corresponding matrix.

Size

Youth

Sweat Pants

Sweat Shirts

Shorts

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T-shirts

The data in the above table can be entered in the shape of a matrix as follows:

DIMENSION

Dimension or Order of a Matrix = Number of Rows x Number of Columns

Example

Matrix T has dimensions of 2x3 or the order of matrix T is 2x3. `×' is just the

notation, it do not mean to multiply both of them.

ROW, COLUMN AND SQUARE MATRIX

Suppose n = 1,2,3,4,.......

A matrix with dimensions 1xn is referred to as a row matrix

For example, matrix A to the right is a 1x4 row matrix.

A matrix with dimensions nx1 is referred to as a column matrix.

For example, matrix B in the middle is a 2x1 column matrix.

A matrix with dimensions nxn is referred to as a square matrix.

For example, matrix C is a 3x3 square matrix.

IDENTITY MATRIX

An identity matrix is a square matrix with 1's on the main diagonal from the upper

left to the lower right and 0's off the main diagonal. An identity matrix is denoted

as I. Some examples of identity matrices are shown below. The subscript

indicates the size of the identity matrix. For example,

, represents an identity

matrix with dimensions n n.

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MULTIPLICATIVE IDENTITY

With real numbers, the number 1 is referred to as a multiplicative identity

because it has the unique property that the product a real number and 1 is that

real number. In other words, 1 is called a multiplicative identity because for any

real

number n, 1 n = n and n 1=n. With matrices, the identity matrix shares the

same unique property as the number 1. In other words, a 2 2 identity matrix is a

multiplicative inverse because for any 2 2 matrix A,

A = A and A

Example

Given the 2 2 matrix, A =

Work

r1c1 = 1(2) + 0(-3) = 2

r2c1 = 0(2) + 1(-3) = -3

r1c1 = 2(1) + -1(0) = 2

r2c1 = -3(1) + 4(0) = -3

r1c2 = 1(-1) + 0(4) = -1

r2c2 = 0(-1) + 1(4) = 4

r1c2 = 2(0) + -1(1) = -1

r2c2 = -3(0) + 4(1) = 4

where 'r' is for row and 'c' is for column.

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