Question

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

Answer 1

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

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

$\Rightarrow (1-\lambda )(2-\lambda )(3-\lambda )-2\left(2\right)(2-\lambda )=0$

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

$3{\lambda }^2-{\lambda }^3+6-2\lambda -9\lambda +3{\lambda }^2-8+4\lambda =0$

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

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

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

$\Rightarrow k=2$