## Matrix Introduction

Matrices are everywhere. If you have used a spreadsheet program such as MS-Excel, Lotus or written a table (such as in Ms-Word) or even have used mathematical or statistical software such a Mathematica, Matlab, Minitab, SAS, SPSS and Eviews etc., you have used a matrix.

Matrices make the presentation of numbers clearer and make calculations easier to program. For example, the matrix is given below about the sale of tires in a particular store given by quarter and make of tires.

 Q1 Q2 Q3 Q4 Tirestone 21 20 3 2 Michigan 5 11 15 24 Copper 6 14 7 28

It is called matrix, as information is stored in particular order and different computations can also be performed. For example, if you want to know how many Michigan tires were sold in Quarter 3, you can go along the row ‘Michigan’ and column ‘Q3’ and find that it is 15.

Similarly, total number sales of ‘Michigan’ tiers can also be found by adding all the elements from Q1 to Q4 in Michigan row. It sums to 55. So, a matrix is a rectangular array of elements. The elements of a matrix can be symbolic expression or numbers. Matrix [A] is denoted by;

Row i of the matrix [A] has n elements and is [ai1, ai2, … a1n] and column of [A] has m elements and is .

The size (order) of any matrix is defined by the number of rows and columns in the matrix. If a matrix [A] has m rows and n columns, the size of the matrix is denoted by (m x n). The matrix [A] can also be denoted by [A]mxn to show that [A] is a matrix that has m rows and n columns in it.

Each entry in the matrix is called the element or entry of the matrix and is denoted by aij, where i represents the row number and j is the column number of the matrix element.

The above-arranged information about sale and type of tires can be denoted by the matrix [A], that is, This matrix has 3 rows and 4 columns. So, the order (size) of the matrix is 3 x 4. Note that element a23 indicate the sales of tires in ‘Michigan’ in quarter 3 (Q3).

## Number System

In early civilizations, the number of animals (sheep, goat, and camel etc.) or children people have were tracked by using different methods such as people match the number of animals with the number of stones. Similarly, they count the number of children with the number of notches tied on a string or marks on a piece of wood, leather or wall. With the development of human, other uses for numerals were found and this led to the invention of the number system.

## Natural Numbers

Natural numbers are used to count the number of subjects or objects. Natural numbers are also called counting numbers. The numbers $1, 2, 3, \cdots$ are all natural numbers.

## Whole Numbers

The numbers $0, 1, 2, \cdots$ are called whole numbers. It can be observed that whole numbers except 0 are natural numbers.

## Number Line

Whole numbers can be represented by points on a line called the number line. For this purpose, a straight line is drawn and a point is chosen on the line and labeled as 0. Starting with 0, mark off equal intervals of any suitable length. Marked points are labeled as $1, 2, \cdots$ as shown in Figure below. The figure below represents real numbers since it includes the negative number (numbers on the left of 0 in this diagram are called negative numbers).

The arrow on the extreme (right-hand side in case of while numbers or negative numbers) indicates that the list of numbers continues in the same way indefinitely.

A whole number can be even or odd. An even number is a number which can be divided by 2 without leaving any remainder. The numbers $0, 2, 4, 6, 8, \cdots$ are all even numbers. An odd number is a number which cannot be divided by 2 without leaving any remainders. The numbers $1, 3, 5, 7, 9, \cdots$ are all odd numbers.

It is interesting to know that any two numbers can be added in any order and it will not affect the results. For example, $3+5 = 5+3$. This is called the commutative law of addition. Similarly, the order of grouping the numbers together does not affect the result. For example, $2+3+5=(2+3)+5 = 2+ (3+5)=(2+5)+3$. This is called the associative law of addition. The subtraction and division of numbers are not commutative as $5-7\ne7-5$ and $6\div2 \ne 2\div 6$ in general.

Like addition and multiplication, whole numbers also follow commutative law and it is called commutative law of multiplication, for example, $2\times 8 = 8 \times 2$. Like addition and multiplication, whole numbers also follow associative law of multiplications. For example, $2 \times (3 \times 4) = (2 \times 3) \times 4 = (2 \times 4)\times 3$. Similarly, multiplication is distributive over addition and subtraction, for example, (i) $5\times (6 + 7) = (5 \time 6) + (5 \times 7)$ or $(6+7) \times 5=(6 \times 5)+(7 \times 5)$. (ii) $3 \times (6-2) = (3 \times 6) - (3 \times 2)$ or $(6-2) \times 3 = (6 \times 3) - (2 \times 3)$.

Take any two digit number say 57, reverse the digits to obtain 75. Now subtract the smaller number from the bigger number, we have $75-57=18$. Now reverse the digits of 18 and add 18 to its reverse (81), that is, 18+81, you will get 99.