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.
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
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.