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.