## Types of number

There are various types of number as follow

## 1. Natural Number

Nature number are counting number and these are denoted by   ,

i.e.,

* All natural number are positive.

* Zero is not a natural number, therefore 1 is the smallest natural number.

## 2. Whole number

All natural number and zero form the set of whole number and these are denoted by W,

i.e,

* Zero is the smallest whole number

* Whole number are also called as non-negative integers.

## 3. Integers

Whole number and negative number form the set of integers and there are denoted by ,

i.e.,

Integers are of following two types

1. Positive integers:- Natural numbers are called as positive integers and these are denoted by  ,

i.e.,

2. Negative integers:- Negative of natural number are called as negative integers and these are denoted by

i.e.,

* 0 is neither +ve nor -ve integers.

## 4. Even number

A counting number which is divisible by 2, is called an even number

For example, 2,4,6,8,10,12,....etc.

* The unit's place of every even number will be 0,2,4,6, or 8.

## 5. Odd number

A counting number which is not divisible by 2, is called an odd number

For example, 1,3,5,7,9,11,13,15,17,....etc.

* The unit's place of every odd number will be 1,3,5,7 or 9.

## 6. Prime number

A counting number is called a prime number when it is exactly divisible by only 1 and itself.

For example 2,3,5,7,11,13,.... etc.

* 2 is the only even number which is prime.

* A prime number is always greater then 1.

* 1 is not a prime number therefore the lowest odd prime number is 3.

* Every prime number greater than 3 can be represented by 6n+1, where n is integer.

## 7. Composite number

Composite number are non-prime natural number. They must have atleast one factor apart from 1 and itself.

For example 4,6,8,9,..etc.

* composite number can be both odd and even.

* 1 is neither a prime number nor a composite number.

## 8. Coprime

Two natural number are said to be  coprime, if their common divisor is 1

For example (7,9), (15,16), etc.

* coprime number may or may not be prime.

* Every pair of consecutive number is coprime.

## 9. Rational number

A number that can be expressed in the form of p/q is called a rational number, where p,q are integers and  .

For example

## 10. Irrational number

The number that cannot be expressed i  form of p/q are called irrational number, where p,q are integers and .

For example

*   is an irrational number as 22/7 is not the actual value of   but it is its nearest value.

* Non-periodic infinite decimal fractions are called irrational numbers.

## 11. Real numbers

Real number  include both rational and irrational number. They are denoted by .

For example

## Cayley Hamilton theorem

Cayley Hamilton theorem state that every square matrix satisfies its characteristic equation.

### Theorem

Proof:-

Let A be any square matrix of order n, and its characteristic equation be

We have to prove that A satisfies this equation

..(1)

For proving this, we proceed as follow :

We know that

Let

We have,

Equating the coefficient of like power of   , we get,

... ... ... ... ...

Pre-multiplying above equation by  respectively and adding, we get,
, which is same as (1).

Hence the theorem

#### Example

Q1. Verify Cayley Hamilton theorem for the matrix

Sol

=

The characteristic equation of A is

We have to prove that A satisfies this equation i.e.,

(1) is satisfied.
Hence the result

## Rank of a matrix

In linear algebra, the rank of a matrix is the dimension of its row space or column space. It is important to note that the row space and column space of a matrix have equal dimensions.

### Definition of rank of a matrix

A nimber r is said to be a rank of a non-zero matrix A if

1. There exist at least one minor of order r of A which does not vanish and

2. Every minor of higher order than r is zero.

The rank of a matrix A is denoted by  .
We have

#### Another definition of rank of a matrix

The rank of a non-zero matrix is largest order of any non-vanishing minor of the matrix.

Remarks

From the above definition of rank of a matrix, we observe that

1. The rank of zero matrix is zero i.e.,   where O is a zero matrix,

2. The rank of a non-singular matrix of order n is n,

3.  , if every minor of order r+1 vanishes,

4.  , if there is a minor of order r which does not vanish.

### ☆Finding rank of matrix

1. Minor method

2. Normal form

3. Echelon form of matrix

#### 1. Example on minor method

Q1. Determine the rank of matrix

Ans

A is non singular

Q2. Determine the rank of matrix

Ans.

i.e, only minor of order 3 of A vanishes.
Now we consider any minor of order 2.

Consider
There is a minor of order 2 of A which does not vanish

#### 2. Normal form of a matrix

are called the normal formes of matrix

#### Example

Q3. Prove that the matrix   is equivalent to  .
Ans

Given matrix is equivalent to  .

#### 3. Echelon form

A matrix   is said to be in echelon form if

1. The zero rows of A occur below all the non-zero rows of A

2. The number of rows before the first non-zero element in a row is less than the number of such zero in the next row.

#### Example

Q4. Reduce to row echelon form the matrix

Ans.

Which is in row echelon form.
Since there are two non-zero rows in row echelon form.

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