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

a_oλⁿ + a₁λ^n-1 + a₂λ^n-2 + ………………. + a_n-2λ² + a_n-1λ + 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_oPⁿ + a₁P^n-1 + a₂P^n-2 + ………………. + a_n-2P² + a_n-1P + a_n I_n= O

(λ is replaced by matrix P in the characteristic equation and a_n replaced by a_nI_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

$P=\begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}$

The characterisitc equation of matrix P is

|P – λI| = 0

$\Rightarrow \begin{vmatrix} 1-\lambda & 4\\ 2 & 3-\lambda \end{vmatrix}=0 \\ \Rightarrow (1-\lambda)(3-\lambda)-8=0 \\ \Rightarrow \lambda^{2}- 4 \lambda -5 = 0$

According to Cayley-Hamilton theorem, we have

⇒ P² – 4P – 5I = O

$\Rightarrow I=\frac{1}{5}[P^{2}-4P]$

Pre-multiplying by P^-1, we get

$P^{-1}=\frac{1}{5}[P-4I] \\ P^{-1}=\frac{1}{5}\left ( \begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}-4\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \right ) \\ P^{-1}=\frac{1}{5}\begin{bmatrix} -3 & 4\\ 2 & -1 \end{bmatrix}$

This is the inverse of matrix P.

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