Chapter 7.1, Problem 49E

Cayley-Hamilton Theorem In Exercises 49-52, demonstrate the Cayley-Hamilton Theorem for the matrix $A$. The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of $A=\left[\begin{array}{cc}1& -3\\ 2& 5\end{array}\right]$is,

${\lambda }^2-6\lambda +11=0$, and by the theorem you have,

$A^2-6A+11I_2=O$.

$A=\left[\begin{array}{cc}5& 0\\ -7& 3\end{array}\right]$