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.