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
\[\begin{equation} \begin{aligned}det(A- \lambda I) &= det(\left( \begin{array}{cc} 2 & 3 \\ 3 & 2 \end{array} \right) - \lambda \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) ) \\ &= det \left( \begin{array}{cc} 2 - \lambda& 3 \\ 3 & 2 - \lambda \end{array} \right) \\ &= (2- \lambda )^2-3^2 \\ &= \lambda^2-4 \lambda -5\end{aligned} \end{equation}\]

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

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