**Cayley-Hamilton theorem** states that *every square matrix satisfies its own characteristic equation*. This theorem is named after two mathematicians, Arthur **Cayley** & William Rowan **Hamilton**. This theorem provides an alternative way to find the inverse of a matrix.

Let

a_{o}λ^{n} + a_{1}λ^{n-1} + a_{2}λ^{n-2} + ………………. + a_{n-2}λ^{2} + a_{n-1}λ + a_{n} = 0

be the characteristic equation of square matrix P of order n. Then, according to the Cayley-Hamilton theorem, matrix P will satisfy this characteristic equation i.e.

a_{o}P^{n} + a_{1}P^{n-1} + a_{2}P^{n-2} + ………………. + a_{n-2}P^{2} + a_{n-1}P + a_{n} I_{n}= O

(λ is replaced by matrix P in the characteristic equation and a_{n} replaced by a_{n}I_{n}, where I_{n} is the identity matrix of order n and O is the null or zero matrix of order n.

## Inverse of a matrix using Cayley-Hamilton theorem

Let us take an example of 2 x 2 matrix P such that

The characterisitc equation of matrix P is

|P – λI| = 0

According to Cayley-Hamilton theorem, we have

⇒ P^{2} – 4P – 5I = O

Pre-multiplying by P^{-1}, we get

This is the inverse of matrix P.