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