(Solved): V. Definition The minimum polynomial of a matrix AMn(R) is the polynomial p(x) of least degre ...

V. Definition The minimum polynomial of a matrix $A?M_n(R)$ is the polynomial $\mathrm{p}(\mathrm{x})$ of least degree such that $p(A)=0$. Theorem Cayley-Hamilton If ${?}_A(x)$ is the characteristic polynomial of $\mathrm{A}$, then ${?}_A(A)=0$. Thus, the minimum polynomial of a matrix divides its characteristic polynomial. Question Find the minimum polynomial of $A=?\begin{array}{ccc}1& ?1& 4\\ 3& 2& ?1\\ 2& 1& ?1\end{array}?$.