The Cayley-Hamilton theorem

The characteristic equation of a square matrix $A$ is given by Eq. (10.90). The CayleyHamilton theorem states that any $n \times n$ matrix $A$ solves its own characteristic equation, that is

where the RHS of Eq. (10.118) is the $n \times n$ zero matrix. Consider the matrix,

The characteristic polynomial is

We now show that

Application of the Cayley-Hamilton theorem to finding inverses

Continuing from the example above, we have shown that

multiplying both sides by $A^{-1}$ gives,

this rearranges to,

Plugging in $A$ in Eq. (10.120),

It is always a good idea to check that $A^{-1} A=I$,

Example 10.12 Find $A^{-1}$ using the Cayley-Hamilton theorem for a matrix $A$ given by,

Solution To use the Cayley-Hamilton theorem, we first need to determine the characteristic equation. With the row transformation $\left(R_{1}-R_{2}\right) \rightarrow\left(R_{1}\right)$ on $A-\lambda I$, we have

By Cayley-Hamilton, we have the following expression in terms of the matrix $A$,

We take $A^{-1}$ on both sides which gives,

and this gives the inverse in the form,

We now need to determine the matrices on the RHS of Eq. (10.124),

This gives us the inverse as,