Cayley-Hamilton Theorem

Definition
Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic equation.
That is, if p ( ) = det ( A - I ) is the characteristic polynomial of the matrix A, then according to Cayley-Hamilton Theorem p ( A ) = 0

For example,
If p ( ) = det ( A - I ) = 2 - 15 + 56 is the characteristic polynomial, then by Cayley-Hamilton Theorem A2 - 15 A + 56 I = 0
The follwing examples will make the matter more clear.


Practice Problems

Example 1. If

then show that A3 = 7 A - 6 I
Solution

the characteristic polynomial of A is

Now, by Cayley-Hamilton Theorem
A2 - 3 A + 2 I = 0 ------> (1)
From (1): A2 = 3 A - 2 I
multiply A on both sides of above equation
A3 = 3 A2 - 2 A -------->(2)
Put A2 = 3 A - 2 I in equation (2)
(2) A3 = 3 ( 3 A - 2 I ) - 2 A
A3 = 9 A - 6 I - 2 A
A3 = 7 A - 6 I Answer

Example 2.If

then show that A5 = 981 A - 540 I
Solution
The characteristic polynomial of A is

Now, by Cayley-Hamilton Theorem
A2 - 6 A + 3 I = 0 ------> (1)
From (1): A2 = 6 A - 3 I
multiply A on both sides of above equation
A3 = 6 A2 - 3 A -------->(2)
Put A2 = 6 A - 3 I in equation (2)
(2) A3 = 6 ( 6 A - 3 I ) - 3 A
A3 = 36 A - 18 I - 3 A
A3 =33 A - 18 I
multiply A2 on both sides of above equation
A5 = 33 A3 - 18 A2 -------->(3)
Put A2 = 6 A - 3 I and A3 =33 A - 18 I in equation (3)
(3) A5 = 33 ( 33 A - 18 I ) - 18 ( 6 A - 3 I )
A5 = 1089 A - 594 I - 108 A + 54 I
A5 = 981 A - 540 I Answer