Cayley-Hamilton Theorem

Examples of Cayley Hamilton Theorem

We will prove the theorem for a 2 x 2 square matrix and a 3 x 3 square matrix.

For 2 x 2 Matrix

Let’s take an example where A is the square matrix and is equal to 

We are now putting the value of A and x as λ in equation 1. We will get the following equation:

Now solving the above determinant will result in the following equation:

To prove the Cayley Hamilton theorem, we need to put A in the above equation in place of λ, and the result must be equal to a zero matrix.

It is coming equal to a zero matrix. Hence it satisfies the Cayley Hamilton Theorem.

For 3 x 3 Matrix

Let’s take a 3 x 3 square matrix A as below:

We will now put the value of A and the value of x as λ in equation 1. By making the following changes, the equation will look like this:

We need to get an equation to prove the Cayley Hamilton theorem that we will get by solving the above determinant. 

Solving the above determinant, the equation will be like this:

To satisfy the Cayley Hamilton theorem, we will substitute A in place of λ in the above equation, so the above equation will convert into the below equation:

Now substituting the value of A into the equation:

The result is a Zero matrix. Hence the Cayley Hamilton theorem is satisfied.

FAQs

1. What do you understand by Cayley Hamilton's theorem?
As per this theorem, all the square matrices satisfy their characteristic equation.
 

2. What is the formula or equation for Cayley Hamilton's theorem?
p(x) = det(xI_n - A) is the equation where A is the square matrix and x is the variable of the polynomial p.
 

3. On what type of matrix is this equation applicable?
This equation applies to square matrices.
 

4. What do you mean by characteristics of a polynomial?
These refer to the polynomial properties that give information about the matrix.

Key Takeaways

In this article, we have extensively discussed the Cayley-Hamilton theorem, its equation, how to find them, and all this with examples to better understand.

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