## The Cayley Hamilton Theorem

The Cayley Hamilton Theory states that a matrix is a root of its own characteristic equation.
Let
$A=\left( \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} \right)$
.
The characteristic equation arises from the equation
$det(A- \lambda I)=0$
. This equation results in a polynomial in terms of
$\lambda$
which thee matrix also satisfies. For the matrix above
\begin{aligned}det(A- \lambda I) &= det(\left( \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} \right) - \lambda \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) ) \\ &= det \left( \begin{array}{cc} 2 - \lambda& 3 \\ 3 & 2 - \lambda \end{array} \right) \\ &= (2- \lambda )^2-3^2 \\ &= \lambda^2-4 \lambda -5\end{aligned}

Substituting
$A$
into this expression gives
\begin{aligned} A^2-4A-5I \\ &= \left( \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} \right)^2- 4 \left( \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} \right) - \left( \begin{array}{cc} 5 & 0 \\ 0 & 5 \end{array} \right) \\ &= \left( \begin{array}{cc} 13 & 12 \\ 13 & 12 \end{array} \right)- \left( \begin{array}{cc} 8 & 12 \\ 12 & 8 \end{array} \right) - \left( \begin{array}{cc} 5 & 0 \\ 0 & 5 \end{array} \right) \\ &= \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right) \end{aligned}