Thursday, 18 April 2019

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.

                       

Wednesday, 27 March 2019

Exact Differential Equations

The differential equation of type
           (where M and N are function of x and y) is called an exact differential equation when
    ( where u is a function of x and y).

Examples :
   
    1.   is an exact differential equation as  

    2.   is an exact differential equation as 

    3. The differential equation 
 is an exact differential equations as 

Necessary and sufficient condition


Article. Find the necessary and sufficient condition that the equation   ( where M and N are function of x and y with the condition that   are continuous function of x and y) may be exact.

Proof  1. Necessary condition
                   

            2. Condition is sufficient

     

Integrating factor


An integrating factor (abbreviatef I.F) of a differential equation is such a factor such that if the equation is multiplied by it, the result equation is exact.

Five rules for finding integrating factor


If   is not exact and it is difficult to find integrating factor, then following five rules help us in finding integrating factor.

Rule 1. If the equation   is homogenous in x and y i.e. if M and N are homogenous function of the same degree in x and y, then   is an I.F. provided 

Rule 2. If the equation   is of the form  , then   is an I.F. provided  

Rule 3. If the equation   is a function of x only =f(x) then   is an I.F.

Rule 4. If the equation   is a function of y only =f(y) then   is an I.F.

Rule 5. If the equation is 
, then   is an I.F. when   .

Example


Q1. Solve the differential equation
      .
Sol. The given differential equation is
 
Comparing it with , we get
         
         
                  
Given equation is exact and its solution is

 

    

     

     

Q2. Solve the following differential equation 
Sol.
      The given differential equation is
        

Which is homogenous in x,y.
Comparing with  , we get

               


Multiple both side by  , we get,



Which is exact and its solution is

       

    

      

       

       

Q3. Solve tge differential equation
  
Sol.
     The given differential equation is

 



Which is of form  

Comparing it with  , we get



I.F.= 
     
=

 =
Multiple both side by , we get



Which is exact and its solution is






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