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Cayley-Hamilton theorem is quite an important theorem used in matrix theory. This theorem is named after its founder mathematicians **Arthur Cayley **and **William Hamilton**.

**Explanation:**

Let us consider that A be an n×nn×n square matrix and if its characteristic polynomial is defined as:

P(λ)=|A−λ I_{n}|

Where, I_{n } is the identity matrix of same order as A .

**Then, according to Cayley-Hamilton theorem:**

P(A) = O

**Where**, O represents the zero matrix of same order as A .

We can say that if we replace λ by matrix A , then the relation would be equal to zero. Hence matrix A annihilates its own characteristic equation.

**Proof**

Let us assume a square matrix A of dimension n×nn×n. If P(λ) be its characteristic polynomial, then by the definition of characteristic polynomial:

Also, let us suppose that Q(λ) be the adjoint matrix of A−λI, such that:

We have the formula:

(adjA)A = (detA)I

Also, we have:

On comparing (1) and (2), we get:

On multiplying ascending powers of A in each equation,

On adding all the equations together, Everything on left hand side cancels out and we obtain,

Hence, the statement of Cayley Hamilton theorem is proved.

**Examples**

**The examples based on Cayley Hamilton theorem are illustrated below:**

**Example 1 : ** Prove Cayley Hamilton theorem for the following matrix?

**Solution: ** Let A =

Let us find characteristic polynomial of given matrix.

In order to prove the statement of Cayley Hamilton theorem for A, we need to show that:

P(A) = O, hence Cayley Hamilton theorem for given matrix A is proved.

**Example 2 :** If Cayley Hamilton theorem holds for the matrix, then find its inverse.

Its characteristic polynomial is -

According to Cayley Hamilton theorem -

Therefore, equation (1) becomes -

**Applications**

Cayley Hamilton theorem is widely applicable in many fields not only related to mathematics, but in other scientific fields too. This theorem is used all over in **linear algebra**. One can easily find **inverse of a matrix **using Cayley Hamilton theorem. It also plays an important role in solving **ordinary differential equations**. This theorem is quite useful in **physics** also. Cayley Hamilton theorem plays a vital role in **computer programming and coding**. In a newer subject - **Rheology**, where behaviour of material is studied, this theorem is used to determine the equations that illustrate nature of materials.

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