# Cayley-Hamilton Theorem

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

aoλn + a1λn-1 + a2λn-2 + ………………. + an-2λ2 + an-1λ + an = 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.

aoPn + a1Pn-1 + a2Pn-2 + ………………. + an-2P2 + an-1P + an In= O

(λ is replaced by matrix P in the characteristic equation and an replaced by anIn, where In 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

⇒ P2 – 4P – 5I = O

Pre-multiplying by P-1, we get

This is the inverse of matrix P.

This site uses Akismet to reduce spam. Learn how your comment data is processed.

error: Content is protected !!