# 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],

A^{2} - (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: A^{n}.

## Example 1

### Example

### Solution

The power of A, A^{3}, is given.

So use the Cayley-Hamilton theorem.

Write A^{2}

minus,

a + d, 2 + 0

A.

plus,

ad - bc, 2⋅0 - 3⋅1

I

is equal to O.

So

A^{2} - (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 A^{2} = 2A + 3I.

See the given equation

A^{3} = 7A + 6I.

Start from the left side A^{3}.

A^{3} = AA^{2}

A^{2} = 2A + 3I

Then

AA^{2} = A(2A + 3I).

Substitution Method

A(2A + 3I)

= 2A^{2} + 3AI

= 2A^{2} + 3A

Multiply a Monomial and a Polynomial

A^{2} = 2A + 3I

Then

2A^{2} + 3A = 2(2A + 3I) + 3A.

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

4A + 3A = 7A

A^{3} = 7A + 6I

So

A^{3} = 7A + 6I

is true.

This is the solution of this example.

## Example 2

### Example

### Solution

The power of A, A^{10}, is given.

So use the Cayley-Hamilton theorem.

Write A^{2}

minus,

a + d, -1 + 1

A.

plus,

ad - bc, (-1)⋅1 - 1⋅1

I

is equal to O.

So

A^{2} - (-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 A^{2} = 2I.

It says to find A^{10}.

To use A^{2} = 2I,

change A^{10} to (A^{2})^{5}.

Power of a Power

A^{2} = 2I

Then

(A^{2})^{5} = (2I)^{5}.

(2I)^{5} = 2^{5}I^{5}

Power of a Product

2^{5} = 32

I^{5} = I

So A^{10} = 32I.