Cayley-Hamilton Theorem (2x2)

How to use the Cayley-Hamilton theorem to simplify the power of a 2x2 matrix: formula, 2 examples, and their solutions.

Formula

Formula

For a matrix A = [a b / c d],
A2 - (a + d)A + (ad - bc)I = O
is true.

I: Identity matrix
O: Zero matrix

This is the Cayley-Hamilton theorem.

The Cayley-Hamilton theorem can be used
to simplify the power of A: An.

Example 1

Example

Solution

The power of A, A3, is given.
So use the Cayley-Hamilton theorem.

Write A2

minus,
a + d, 2 + 0
A.

plus,
ad - bc, 2⋅0 - 3⋅1
I

is equal to O.

So
A2 - (2 + 0)A + (2⋅0 - 3⋅1)I = O.

-(2 + 0)A = -2A
+(2⋅0 - 3⋅1)I = +(0 - 3)I

+(0 - 3)I = -3I

Move -2A - 3I to the right side.

Then A2 = 2A + 3I.

See the given equation
A3 = 7A + 6I.

Start from the left side A3.

A3 = AA2

A2 = 2A + 3I

Then
AA2 = A(2A + 3I).

Substitution Method

A(2A + 3I)
= 2A2 + 3AI
= 2A2 + 3A

Multiply a Monomial and a Polynomial

A2 = 2A + 3I

Then
2A2 + 3A = 2(2A + 3I) + 3A.

2(2A + 3I) = 4A + 6I

4A + 3A = 7A

A3 = 7A + 6I

So
A3 = 7A + 6I
is true.

This is the solution of this example.

Example 2

Example

Solution

The power of A, A10, is given.
So use the Cayley-Hamilton theorem.

Write A2

minus,
a + d, -1 + 1
A.

plus,
ad - bc, (-1)⋅1 - 1⋅1
I

is equal to O.

So
A2 - (-1 + 1)A + ((-1)⋅1 - 1⋅1)I = O.

-(-1 + 1)A = -0A
+((-1)⋅1 - 1⋅1)I = +(-1 - 1)I

+(-1 - 1)I = -2I

Move -2I to the right side.

Then A2 = 2I.

It says to find A10.

To use A2 = 2I,
change A10 to (A2)5.

Power of a Power

A2 = 2I

Then
(A2)5 = (2I)5.

(2I)5 = 25I5

Power of a Product

25 = 32
I5 = I

So A10 = 32I.