Question

If $A=\left[\begin{array}{ccc}1& 2& 3\\ 3& -2& 1\\ 4& 2& 1\end{array}\right],$ then show that $A^3-23A-40I=0$.

Answer 1

Given $A=\left[\begin{array}{ccc}1& 2& 3\\ 3& -2& 1\\ 4& 2& 1\end{array}\right]$

$|A-\lambda I|=\left|\begin{array}{ccc}1-\lambda & 2& 3\\ 3& -2-\lambda & 1\\ 4& 2& 1-\lambda \end{array}\right|=0$

$(1-\lambda )({\lambda }^2+\lambda -4)-2(7-3\lambda )+3(4\lambda +14)=0$\\ ${\lambda }^2+\lambda -4-{\lambda }^3-{\lambda }^2-4\lambda -14+6\lambda +12\lambda +42=0$

${\lambda }^3-23\lambda -34=0$

Characterstic equation of $A$ is $A^3-23A-34I=0$