Question

If $A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]$ and $A^3-6A^2+7A+kI=O$, find the value of k.

Answer 1

$A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right]$

$|A-\lambda I|=0$

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

$(1-\lambda )(2-\lambda )(3-\lambda )+2(-2)(2-\lambda )=0$

$(2-\lambda )[{\lambda }^2-4\lambda +3-4]=0$

$(2-\lambda )[{\lambda }^2-4\lambda -1]=0$

$2{\lambda }^2-8\lambda -2-{\lambda }^3+4{\lambda }^2+\lambda =0$

$\Rightarrow {\lambda }^3-6{\lambda }^2+7\lambda +2=0$

$\Rightarrow A^3-6A^2+7A+2I=0$

$\therefore k=2$.